Figure 2.3 Velocity Profile for Laminar vs Turbulent Flows . .... Figure 4.9 Integrated RANS-LES Computations in Gas Turbines: Compressor-Diffuser, ............. 49 ...... 24 Training Manual, ANSYS CFX. ...... 52 T. Ganesan and M. Awang, âLarge Eddy Simulation (LES) for Steady-State Turbulent Flow Predictionâ, Springer.
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CFD Open Series Revision 1.85.4
Turbulence Modeling A Review Ideen Sadrehaghighi, Ph.D.
Turbulence contemporary abstaract By Nancy Eckels
ANNAPOLIS, MD
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Contents 1
Introduction .................................................................................................................................. 7
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Turbulence Essentials ............................................................................................................... 9 2.1 Physical Perspectives ........................................................................................................................................... 9 2.2 Components Attributing to Complexity of Physics in Turbulence .................................................... 9 2.2.1 Enhanced Diffusion ................................................................................................................................ 10 2.2.2 Stability Criteria ...................................................................................................................................... 10 2.2.3 Time Dependency ................................................................................................................................... 10 2.2.4 Eddies and Spectral Length ................................................................................................................ 11 2.2.5 Non-Linearity Effects ............................................................................................................................ 11 2.2.6 Separation and Drag Reduction ........................................................................................................ 12 2.3 Transition ............................................................................................................................................................... 12
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Reynolds Averaging and RANS Modeling (Linear Modeling) .................................... 15
3.1 Time Averaging .................................................................................................................................................... 15 3.1.1 Un-Steady RANS (URANS)................................................................................................................... 16 3.2 Reynolds Averaged Naiver-Stokes (RANS) .............................................................................................. 16 3.2.1 Comments on Reynolds Averaged Naiver-Stokes (RANS) ..................................................... 17 3.3 Closure Problem (Boussinesq Assumption) ............................................................................................ 18 3.4 Turbulence Models ............................................................................................................................................. 18 3.4.1 Turbulent Viscosity, Velocity, and Length Scale ........................................................................ 20 3.4.1.1 Eddy Viscosity (RANS) Models ................................................................................. 20 3.4.1.2 Algebraic Model (Zero Equation) - Mixing Length .................................................. 21 3.4.1.3 Baldwin-Lomax ........................................................................................................ 21 3.4.2 One-Equation Model .............................................................................................................................. 21 3.4.2.1 k - Model.................................................................................................................. 22 3.4.2.2 Spallart - Allmaras.................................................................................................... 22 3.4.3 Two Equation Model ............................................................................................................................. 22 3.4.3.1 Standard Transport Equation for k .......................................................................... 23 3.4.3.2 Standard Transport Equation for ε .......................................................................... 24 3.4.3.3 The RNG Model ....................................................................................................... 25 3.4.3.4 The Realizable κ-ε Model ........................................................................................ 25 3.4.3.5 Issues Relating to κ-ε Turbulence Modeling............................................................ 26 3.4.3.6 The κ-ω and SST Models .......................................................................................... 26 3.4.3.7 Wilcox (Standard) κ-ω Model .................................................................................. 27 3.4.3.8 The Baseline (BSL) κ-ω Model ................................................................................. 27 3.4.3.9 SST κ-ω Model ......................................................................................................... 27 3.4.4 Case Study – Subsonic Flow for Turbulence Models of (NACA)-0012 Airfoil................ 29 3.4.4.1 Computational Method ........................................................................................... 29 3.4.4.2 Results and Discussion ............................................................................................ 30 3.4.4.3 Conclusions.............................................................................................................. 33 3.4.5 Near Wall Turbulence Modeling ....................................................................................................... 33 3.4.5.1 Modeling Flow Near the Wall .................................................................................. 34 3.4.5.2 Case Study - Wall-modelling strategies in Large Eddy Simulation of separated highReynolds-number flows ............................................................................................................... 35
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Beyond the Boussinesq Approximation (Non-Linear Modeling) ............................. 37
4.1 Boussinesq-Approximation Deficiencies .................................................................................................. 37 4.2 Nonlinear Constitutive Relations ................................................................................................................. 37 4.2.1 Case Study – 3D Simulation of Flow Past a Cylinder using Nonlinear Turbulence Model 38 4.2.1.1 Literature Survey ..................................................................................................... 39 4.2.1.2 Basic Equations ........................................................................................................ 39 4.2.1.3 Numerical Strategies ............................................................................................... 40 4.2.1.4 Geometry and Meshing ........................................................................................... 41 4.2.1.5 Results and Discussion ............................................................................................ 41 4.2.1.6 Conclusions.............................................................................................................. 42 nd 4.3 2 Order Closures (Reynolds Stress Models - RSM) ............................................................................ 43 4.4 LES Model ............................................................................................................................................................... 43 4.4.1 Filter Definition ....................................................................................................................................... 44 4.4.2 Filtered Incompressible N-S Equations ......................................................................................... 44 4.4.3 Numerical Methods for LES ................................................................................................................ 46 4.4.4 Implicit vs Explicit Filtering ............................................................................................................... 46 4.4.5 Quantitative Aspects of Comparison of RANS vs. LES Models ............................................. 46 4.4.6 Motivations for Coupling Methods Between RANS and LES ................................................. 47 4.4.6.1 Principal Approaches to Coupling LES with RANS ................................................... 48 4.4.6.2 Results and Discussion for Segregated Modeling ................................................... 49 4.4.6.3 Coupling in RANS/LES Interface............................................................................... 50 4.5 DES Model .............................................................................................................................................................. 51 4.6 DNS Model .............................................................................................................................................................. 52 4.6.1 The Reynolds Number Constraint ................................................................................................... 52 4.6.2 Numerical Considerations ................................................................................................................... 53 4.6.2.1 Spectral Methods .................................................................................................... 54 4.6.2.2 Finite Differencing ................................................................................................... 54 4.7 Strategies for Turbulence Modelling........................................................................................................... 55 4.7.1 Physical Aspects (Modification to Simple RANS Models) ...................................................... 55 4.7.2 Complex RANS Models.......................................................................................................................... 55 4.7.3 Role of Grid Refinement ....................................................................................................................... 56 4.7.4 Outlook ........................................................................................................................................................ 57 4.8 Modeling To Choose From............................................................................................................................... 58
Classification of Turbulence Models .................................................................................. 59 5.1 Comparisons of various Turbulence Models ........................................................................................... 60 5.2 Numerical Considerations ............................................................................................................................... 61 5.2.1 Elementary Time-Marching Methods............................................................................................. 61 5.2.1.1 Block-Implicit Methods ........................................................................................... 61
List of Tables: Table 3.1 Table 3.2 Table 4.1 Table 5.1
Constant values for κ-ε turbulence model equations ............................................................... 24 Constant values for SST κ-ω turbulence model equations..................................................... 28 Summery of Turbulence Strategies .................................... Error! Bookmark not defined. Advantages & Disadvantages of Different Turbulence Models ............................................ 60
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List of Figures:
Figure 2.1 Flow Pattern Pass a Circle..................................................................................................................... 9 Figure 2.2 Large Eddies in a Turbulent Boundary Layer ............................................................................ 11 Figure 2.3 Velocity Profile for Laminar vs Turbulent Flows ..................................................................... 12 Figure 2.4 Turbulence Transition Region ......................................................................................................... 12 Figure 2.5 Snapshots of the Stream-Wise Velocity Component for (A) Adiabatic Wall , (B) Heated Wall, and (C) Cooled Wall ............................................................................................................................. 13 Figure 3.1 Relationship between averaged variables; a) Steady flow b) Un-Steady flow ...... 16 Figure 3.2 Hierarchy of Turbulence Models in General .............................................................................. 19 Figure 3.3 Hierchey of Turbulence Models based on Relative Importance of Numerics vs. Computational Costs - (Courtsy of Tenzor) .......................................................................................................... 19 Figure 3.4 Lift Coefficient at Stall (AoA) against Number of Cells........................................................... 29 Figure 3.5 Comparison Between Experimental Data [Abbott et al.] and Three Different Turbulent Models Simulation Results for Lift Coefficient in NACA 0012 Airfoil .................................. 30 Figure 3.6 Comparison Between Experimental Data for Transitional Boundary Layer and Different Turbulent Models on the Drag Coefficient of NACA-0012 Airfoil ........................................... 31 Figure 3.7 Contours of velocity magnitude at 9° (Top) and 16° (Bottom) AoA with the SpalartAllmaras turbulence model ......................................................................................................................................... 32 Figure 3.8 Zones in Turbulent B. L. for a typical Incompressible flow over a smooth flat plate 33 Figure 3.9 Turbulence and near Wall Function .............................................................................................. 34 Figure 3.10 Schematics of hybrid LES-RANS scheme (upper) and two-layer zonal scheme (lower) ................................................................................................................................................................................. 35 Figure 4.1 Schematic Representation of the Computational Domain ................................................... 41 Figure 4.2 Comparison of Wall Pressure Coefficient (Cp) .......................................................................... 42 Figure 4.3 Streamline Patterns at Four Different Time Intervals for one Vortex Shedding Cycle ...................................................................................................................................................................................... 42 Figure 4.4 Variation of lift Coefficient with Time ........................................................................................... 41 Figure 4.5 Difference Between the Filtered Velocity and the Instantaneous Velocity .................. 43 Figure 4.6 A Velocity Field Produced by a (LES) of Homogeneous Decaying Turbulence....Error! Bookmark not defined. Figure 4.7 Vorticity Prompted by the Wake Passing Cycle ........................................................................ 46 Figure 4.8 Time Averaged (RANS) vs Instantaneous (DES) Simulation Over a Backup Step ...... 47 Figure 4.9 Integrated RANS-LES Computations in Gas Turbines: Compressor-Diffuser, ............. 49 Figure 4.10 Compressor and combustor: RANS and LES axial velocity, mid-passage (Courtesy of Medic et al.) ................................................................................................................................................................... 50 Figure 4.11 DES and Effect of Grid Density on the Wake Flow ............................................................... 51 Figure 4.12 Comparison of Pressure Contours on Planner Cuts for RANS and DES Models ..................................................................................................................... Error! Bookmark not defined. Figure 4.13 Vorticity contours from spectral DNS at two Reynolds numbers (Spalart et al. 2008). ................................................................................................................................................................................... 53 Figure 4.14 The sound radiated by a Mach 1.9 circular jet ........................................................................ 53 Figure 4.15 Instantaneous Contours of Stream-Wise Density Gradients from DNS ....................... 54 Figure 4.16 Stream Lines in Square Channel with Nonlinear Constitutive –( Courtesy of [speziale, 1987]) .............................................................................................................................................................. 55 Figure 4.17 Simulation of Flow Past Circular Cylinder by Various Approaches (Shur et al., 1996; Travin et al., 2000) ............................................................................................................................................. 56
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Preface This note is intended for all undergraduate, graduate, and scholars of Turbulence. It is not completed and never claims to be as such. Therefore, all the comments are greatly appreciated. In assembling this, I was influenced with sources from my textbooks, papers, and materials that I deemed to be important. At best, it could be used as a reference. I also would like to express my appreciation to several people who have given thoughts and time to the development of this article. Special thanks should be forwarded to the authors whose papers seemed relevant to topics, and consequently, it appears here©. Finally I would like to thank my wife, Sudabeh for her understanding and the hours she relinquished to me. Their continuous support and encouragement are greatly appreciated. Ideen Sadrehaghighi June 2018
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1 Introduction Turbulence has been the victim of many colorful descriptions over the years, from Lamb’s (1916) scholarly “chief outstanding difficulty of our subject” to Bradshaw’s (1994) inspired “invention of the Devil on the seventh day of creation.” This apparent frustration results largely from the mixture of chaos and order and the wide range of length and time scales that turbulent flows possess1. The three key elements of CFD are algorithm development, grid generation and turbulence modelling. Turbulence is inherently three-dimensional and time dependent, and an enormous amount of information is thus required to completely describe a Introduction Turbulent Flow. This is beyond the capability of the existing computers for virtually all practical flows. Thus, some kind of approximate and statistical method, called a turbulence model, is needed. Complexity of different turbulence models may vary strongly depends on the details one wants to observe and investigate by carrying out such numerical simulations. N-S equation is inherently nonlinear, time-dependent, three-dimensional PDE. Turbulence could be thought of as instability of laminar flow that occurs at high Reynolds numbers. Such instabilities origin form interactions between non-linear inertial terms and viscous terms in N-S equation. These interactions are rotational, fully time-dependent and fully three-dimensional. Rotational and three-dimensional interactions are mutually connected via vortex stretching. Vortex stretching is not possible in two dimensional space. That is also why no satisfactory two-dimensional approximations for turbulent phenomena are available. Furthermore turbulence is thought of as random process in time. Therefore no deterministic approach is possible. Certain properties could be learned about turbulence using statistical methods. These introduce certain correlation functions among flow variables. However it is impossible to determine these correlations in advance2. Another important feature of a turbulent flow is that vortex structures move along the flow. Their lifetime is usually very long. Hence certain turbulent quantities cannot be specified as local. This simply means that upstream history of the flow is also important of great importance. In short, the turbulence is severely restriction the calculation of CFD in aerospace design process where the inability to reliably predict turbulent flows with significant regions of separation3. Presently, turbulence modelling based on Reynolds-Averaged Navier Stokes (RANS) equations is the most common and practical approach for turbulence simulation. RANS are time-averaged modification of Navier-Stokes equations and turbulence models are semi-empirical mathematical relations that are used to predict the general effect of turbulence. The objective of turbulence modelling is to develop equations that will predict the time-averaged velocity, pressure, and temperature fields without calculating the complete turbulent flow pattern as a function of time. Unfortunately, there is no single universally accepted turbulence model that works for all flows and all regimes. Therefore, users have to use engineering judgement to choose from a number of different alternatives sine the accuracy and effectiveness of each model varies depending on the application.
1 P., Moin and K., Mahesh,
“Direct Numerical Simulation: A Tool in Turbulence Research”, Annual Rev. Fluid Mech. 1998. 30:539–78. 2 J., SODJA, “Turbulence models in CFD”, University of Ljubljana, Faculty for mathematics and physics, Department of physics, March 2007. 3 Dimitri Mavriplis, “Exa-scale Opportunities for Aerospace Engineering“, Department of Mechanical Engineering University of Wyoming and the Vision CFD2030 Team.
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2 Turbulence Essentials Physical Perspectives While the fluid elements are smooth and regular for Laminar flow pass a circle as shown in Figure 2.1 (a), the Turbulent elements are irregular in which various quantities show a random variation with time and space as in Figure 2.1 (b). Therefore, a statically distinct averaging of values can be distinguished. Also, because of this agitated motion in turbulent flow, the higher energy fluid elements from the outer region of the flow are pumped close to surface. The diffusion rate of a scalar quantity is usually greater in a turbulent flow than in a laminar. As the result, the frictional effects are more severe for a turbulent flow. An outstanding feature of turbulent flow, as opposite to laminar flow, is that molecules move in chaotic fashion along complex irregular path. The strong chaotic motion causes the various layers of fluid to mix together intensively. Because of increase momentum and energy exchange between the molecules and solid walls, turbulent flow leads at some conditions to higher skin friction and heat transfer as compared to laminar case. It could be argued as that because of the agitated motion in turbulent flow, the higher energy fluid elements from the outer regions of flow are pumped close to surface, and hence, average flow velocity near solid surface is larger for turbulent flow in comparison to laminar as depicted in following pages as Figure 2.3.
(b) Turbulent flow
(a) Laminar flow Figure 2.1
Flow Pattern Pass a Circle
Components Attributing to Complexity of Physics in Turbulence Before plunging into the mathematics of turbulence, it is worthwhile to first discuss physical aspects of the phenomenon. It is a rather interesting discussion which encompasses most of the physical flow. It is intended for interest parties with full text is available in book by4. The following discussion is not intended as a complete description of this complex topic. Rather, we focus upon a few features of interest in engineering applications, and in construction of a mathematical model. In 1937, Taylor and von Karman proposed the following definition of turbulence: "Turbulence is an irregular motion which in general makes its appearance in fluids, gaseous or liquid, when they flow past solid surfaces (boundary conditions) or even when neighboring streams of the same fluid flow past or over one another (i,e. Jet flow)." It is characterized by the presence of a large range of excited length and time scales. 4
David C. Wilcox, ”Turbulence Modeling for CFD”, 1993, 1994 by DCW Industries, Inc.
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The irregular nature of turbulence stands in contrast to laminar motion, because the fluid was imagined to flow in smooth laminae, or layers. Virtually all flows of practical engineering interest are turbulent. Turbulent flows always occur when the Reynolds number is large. For slightly viscous fluids such as water and air, large Reynolds number corresponds to anything stronger than a small swirl or a puff of wind. Careful analysis of solutions to the Navier-Stokes equation, or more typically to its boundary-layer form, show that turbulence develops as an instability of laminar flow. The features contributing to complexity of turbulence flow are;
Enhance Diffusion Stability Criteria Time Dependency Eddies and Spectral Length Non-Linearity effects Separation
Enhanced Diffusion Perhaps the most important feature of turbulence from an engineering point of view is its enhanced diffusivity. Turbulent diffusion greatly enhances the transfer of mass, momentum and energy. Apparent stresses often develop in turbulent flows that are several orders of magnitude larger than in corresponding laminar flows. Stability Criteria To analyze the stability of laminar flows, virtually all methods begin by linearizing the equations of motion. Although some degree of success can be achieved in predicting the onset of instabilities that ultimately lead to turbulence with linear theories, the inherent nonlinearity of the Navier-Stokes equation precludes a complete analytical description of the actual transition process, let alone the fully-turbulent state. For a real (i.e., viscous) fluid, the instabilities result from interaction between the Navier-Stokes equation's nonlinear inertial terms and viscous terms. The interaction is very complex because it is rotational, fully 3-D and time dependent. The strongly rotational nature of turbulence goes hand-in-hand with its three dimensionality. Vigorous stretching of vortex lines is required to maintain the ever-present fluctuating vorticity in a turbulent flow. Vortex stretching is absent in two-dimensional flows so that turbulence must be three dimensional. This inherent 3-D means there are no satisfactory 2-D approximations and this is one of the reasons turbulence remains the most noteworthy unsolved scientific problem of the twentieth century. Time Dependency The time-dependent nature of turbulence also contributes to its intractability. The additional complexity goes beyond the introduction of an additional dimension. Turbulence is characterized by random fluctuations thus obviating a deterministic approach to the problem. Rather, we must use statistical methods. On the one hand, this aspect is not really a problem from the engineer's view. Even if we had a complete time history of a turbulent flow, we would usually integrate the flow properties of interest over time to extract time-averages. On the other hand, time averaging operations lead to statistical correlations in the equations of motion that cannot be determined a priori. This is the classical closure problem, which is the primary focus of this text. In principle, the time-dependent, three-dimensional Navier-Stokes equation contains all of the physics of a given turbulent flow. That this is true follows from the fact that turbulence is a continuum phenomenon. As noted, "Even the smallest scales occurring in a turbulent flow are ordinarily far larger than any molecular length scale”. Nevertheless, the smallest scales of turbulence are still extremely small. They are generally many orders of magnitude smaller than the largest scales of turbulence, the latter
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being of the same order of magnitude as the dimension of the object about which the fluid is flowing. Furthermore, the ratio of smallest to largest scales decreases rapidly as the Reynolds number increases. To make an accurate numerical simulation (i.e., a full time dependent 3D solution) of a turbulent flow, all physically relevant scales must be resolved. While more and more progress is being made with such simulations, computers of the early 1990's have insufficient memory and speed to solve any turbulent flow problem of practical interest. Eddies and Spectral Length Turbulence consists of a continuous spectrum of scales ranging from largest to smallest, as opposed to a discrete set of scales. In order to visualize a turbulent flow with a spectrum of scales we often refer to turbulent eddies. A turbulent eddy can be thought of as a local swirling motion whose characteristic dimension is the local turbulence scale. Figure 2.2 shows as the flow above the boundary layer has a steady velocity U; the eddies move at randomly fluctuating velocities of the order of a tenth of U. The largest eddy size (l) is comparable to the boundary-layer thickness (δ). The interface and the flow above the boundary is quite sharp [Corrsin and Kistler (1954)]. Eddies overlap in space, large ones carrying smaller ones. Turbulence features a cascading process whereby, as the Figure 2.2 Large Eddies in a Turbulent Boundary Layer turbulence decays, its kinetic energy transfers from larger eddies to smaller eddy. Ultimately, the smallest eddies dissipate into heat through the action of molecular viscosity. Thus, we observe that turbulent flows are always dissipative. An especially striking feature of a turbulent shear flow is the way large bodies of fluid migrate across the flow, carrying smaller-scale disturbances with them. The arrival of these large eddies near the interface between the turbulent region and non-turbulent fluid distorts the interface into a highly convoluted shape. In addition to migrating across the flow, they have a lifetime so long that they persist for distances as much as 30 times the width of the flow [Bradshaw (1972)]. Hence, the turbulent stresses at a given position depend upon upstream history and cannot be uniquely specified in terms of the local strain-rate tensor as in laminar flow. Non-Linearity Effects The nonlinearity of the Navier-Stokes equation leads to interactions between fluctuations of differing wavelengths and directions. As discussed above, the wavelengths of the motion usually extend all the way from a maximum comparable to the width of the flow to a minimum fixed by viscous dissipation of energy. The main physical process that spreads the motion over a wide range of wavelengths is vortex stretching. The turbulence gains energy if the vortex elements are primarily oriented in a direction in which the mean velocity gradients can stretch them. Most importantly, wavelengths that are not too small compared to the mean-flow width interact most strongly with the mean flow. Consequently, the larger-scale turbulent motion carries most of the energy and is mainly responsible for the enhanced diffusivity and attending stresses. In turn, the larger eddies randomly stretch the vortex elements that comprise the smaller eddies, cascading energy to them5.
5
David C. Wilcox, “Turbulence Modeling for CFD”, Copyright © 1993, 1994 by DCW Industries.
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Separation and Drag Reduction Although the frictional effects are more severe in turbulent flow, both shear stress and aerodynamic heating are larger for the turbulent flow in comparison to laminar6. However, turbulent flow has a major redeeming value; because the energy of the fluid elements close to surface is larger, it does not separates from the surface as readily as a laminar flow. Simply, if a flow is turbulent, it is less likely to separate from the body surface. And if flow separates, the separation region is smaller than those for laminar. As the result the pressure drag Dp will be smaller in turbulent flow. This leads to great compromise in aerodynamics. In general, if the body is slender, the friction drag, Df is much greater than Dp. For this case since Df is smaller in laminar flow (lower velocity → lower viscous drag; see Figure 2.3), then laminar flow is desirable for slender bodies (airfoils). In contrast, for a blunt body, Dp is much greater than Df. For this case, since Dp is smaller in turbulent flow, then turbulent flow is desirable (gulf balls).
Figure 2.3
Velocity Profile for Laminar vs Turbulent Flows
Transition The transition from laminar to turbulent is not subtle and consists of several processes, each subject to intense research interests on their own right. The transition and stability of the laminar flow seem to be dependent on a critical value of Reynolds number (Figure 2.4) as
Re cr
Figure 2.4 6
ρ V x cr 2100 μ
Turbulence Transition Region
Anderson, John D. 1984: “Fundamentals of Aerodynamics”, McGraw Hills Inc.
Eq. 2.1
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The equations governing a turbulent flow are precisely the same as for laminar flow; however, solutions is clearly more complicated in this region. That is due to the introduction of new terms and issues with closure. Two general approach could be envisioned. First a more direct approach with extreme spatial discretization to capture all the flow eddies near the wall. This is prohibitly CPU intensive, even with current computing powers. To counter that, the use of some modeling (empirical or otherwise) in the vicinity of wall region is advocated. Rest of this report organized as subsequent. Since subject of Turbulence is very involved, therefore, the physical aspects of formulation, namely Reynolds Stress formulation, and how to derive them represented first. Followed by different aspects of Turbulence modeling. The numerous factors contributing to transition from laminar to turbulent flow in a fluid. The intent has been to provide general background information on the various transition phenomena rather than to make a study of the problem in depth. Included are the effects on transition of such factors as pressure gradient, surface temperature, Mach number, and two- and three-dimensional types of surface roughness [A. L, Braslow]7. The effect of wall heat transfer was investigated by [Shadloo and Hadjadj]8 through high-resolution DNS , among other factors. As in the case of adiabatic wall (Figure 2.5-a), stream-wise-elongated streaks are visible in the laminar and transitional regions for both heated (Figure 2.5-b) and cooled cases (Figure 2.5-c). However, they locally break down and create turbulent spots further downstream for the heated case when compared with the adiabatic wall. Therefore, the wall heating stabilizes the flow and postpones the transition.
(A) Adiadatic Wall (No Heating)
(B) Tw/Tr = 1.5 (Heated)
(C) Tw/Tr = 0.75 (Cooled)
Figure 2.5
Snapshots of the Stream-Wise Velocity Component for (A) Adiabatic Wall , (B) Heated Wall, and (C) Cooled Wall
Albert L, Braslow, “A Review Of Factors Affecting Boundary-Layer Transition”, NASA TN D-3384. M. S. Shadloo & A. Hadjadj, “Laminar-turbulent transition in supersonic boundary layers with surface heat transfer: A numerical study”, Numerical Heat Transfer, 2017. 7 8
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3 Reynolds Averaging and RANS Modeling (Linear Modeling) Time Averaging Since Most of engineers does not concern with instantaneous values but rather deal with averages. Therefore, the Reynolds Averaged N-S equations (RANS) are derived by decomposing the dependent variables in conservative equations into time mean and fluctuating components. Two types of averaging are greatly used, the classical Time (Reynolds) average, and the Mass Weighted average. For flows in which density fluctuating could be neglected, the two formulations becomes identical. The time averaging procedure defined by:
1 f Δt
t 0 Δt
f dt
and f 0 , u i u i ui , ρ ρ ρ , p p p , T T T
Eq. 3.1
t0
It is required that Δt be large compared to the period of random fluctuations associated with turbulence, but small with respect to the time constant for any slow variations in flow field associated with ordinary unsteady flows. In conventional Reynolds decomposition, the randomly changing flow variations are replaced by the time average plus fluctuations (see Figure 3.1) and would be discuss later for Unsteady RANS (URANS) formulations. The resulting of decomposition of variables and time averaging the entire equation, the continuity, momentum, and energy yield to
M ass : M omentum :
ρ (ρ u j ρu j ) t x j (ρ u i ρu i ) (ρ u i u j u i ρu j ) t x j
Energy :
p ( τ u ρu ρ u u ρu u ) ij j i i j i j x x i j
(c ρT c ρ T) p p ρc u T t x p j j
p p p T uj u j k ρc p Tu j c p ρTu j t x j x j x j x j where
τij
ui u τij i x j x j
u u j 2 u k δ ij τij μ i x j x i 3 x k Eq. 3.2
and
1 δ ij 0
i j i j
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Figure 3.1
Relationship between averaged variables; a) Steady flow b) Un-Steady flow
Un-Steady RANS (URANS) For unsteady analysis, the Unsteady RANS could be used where of course the unsteady term is retained in averaging as well. In essence, there are now two time scale involved where the Reynolds decomposition of U would be as before an average and fluctuated value as
U
1
U( ) d
,
U U u
Eq. 3.3
0
Note that the dependent variables are now not only a function of space but also a function of time as well. Be advised that this concept of time, is different from the time step (t) of mean value. In essence if you solve your equations with one global time step, which is used in every cell, and if value of the time step is small enough then you will be able to capture fluctuations, or unsteady behavior in the MEAN quantities. In other words your solution is time accurate. Steady RANS, or RANS, marches the solution with a local optimized time step for each cell, and hence is not time accurate, you will get a faster solution, and it will be steady state. For URANS we have:
Ui Ui (x, y,z, ) and uiuj uiuj (x, y,z, )
Eq. 3.4
Even if the results from URANS are unsteady, one is often interested only in time averaged flow as denoted as , which means that we can decompose the results from URANS as a time averaged part , a resolved fluctuation uˈ, and the modeled turbulent fluctuation, u̎ , i.e.
U U u U u u
Eq. 3.5
Reynolds Averaged Naiver-Stokes (RANS) The Reynolds Averaged Navier-Stokes equations (RANS) are obtained from the continuity and momentum equations by taking the time average of all the terms in the equations. The continuity equation does not change since it is linear in terms of the velocity. However, the momentum equation is non-linear, which means that all the fluctuating components do not vanish. An extra term, called Reynolds stress uiuj, appears in the momentum equation. The result can be written where the
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overbar has been dropped from the mean values. This convention will be used throughout here for obvious reasons. Eq. 3.6 presents the fundamental problem of turbulence. In order to compute all the mean-flow properties of the turbulent flow we need a reasonably accurate way to compute the Reynolds stress uʹiuʹj. This is the fundamental reason for the need of the turbulence models. The complete set of the Reynolds Averaged Navier-Stokes (RANS) equations are not presented here because they can be found from many references9. Scalar transport equations are also needed, for example to describe the transport of the concentration of species or the mass fraction of species. Their exact formulation can be found in10. For incompressible flow with constant properties and no body force the momentum and energy.
Closure Problem
Eq. 3.6 Comments on Reynolds Averaged Naiver-Stokes (RANS) At the first glance the Reynolds equations seemed to be quite complicated to solve in turbulent flow. Certainly a major problem in fluid mechanic is that they are more equation can be written than solved. Fortunately, for many important flows, the Reynolds equations can be simplified. But before turning into the task of simplification, it is prudent to examine the equation themselves. By considering the incompressible flow, the momentum equation can be written as:
ρ
Du̅i ⏟ Dt
= −
Particle Acceleration of Mean Motion
where (τ̅ij )
∂p̅ ∂x ⏟i Mean Pressure Gradient
Turb
′ u′ ̅̅̅̅̅ = −ρu i j
+
∂(τ̅ij )Lam ⏟ ∂xj Laminar like Stress Gradient
+
∂(τ̅ij )Turb ⏟ ∂xj Apperant Stree Gradient due to Transport of Momentum by Turbulent Fluctuation
Similarly for Energy Equation:
P. Kaurinkoski and A. Hellsten, FINFLO: The Parallel Multi-Block Flow Solver, Report A-17, Laboratory of Aerodynamics, Helsinki University of Technology, Espoo, Finland, 1998. 10 P. Kaurinkoski, “Development of an Equation of State for an Arbitrary Mixture of Thermally Perfect Gases to the FINFLO flow solver”, Report No B-48, Series B, Laboratory of Aerodynamics, Helsinki University of Technology, Finland, 1995. 9
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−(∇. q)Lam =
̅ ∂ ∂T (k ) ∂xj ∂xj
and
− (∇. q)Turb =
∂ T ′ u′j ) (ρCp ̅̅̅̅̅̅ ∂xj
Eq. 3.7 The Reynolds equation cannot be solved in the form given because the new apparent turbulent stresses and heat-flux quantities must be viewed as new unknowns. To proceed further, we need to find additional equations involving the new unknowns to make assumption regarding the relationship between the new apparent turbulence quantities and time-mean flow variables. This is known as closure problem which is most commonly handled through turbulence modeling to be discussed next.
Closure Problem (Boussinesq Assumption) A linear relation between turbulent shearing stresses and rate of mean strain suggested by [Boussinesq] which is the fundamental closure for most eddy viscosity models been used.
u i u j 2 u k δij μ T ρui uj μ T ρ κ x x 3 x i k j
κ ui ui /2
Eq. 3.8
Where μT is the turbulent viscosity, κ is the kinematic energy of turbulence are key modeling aspects associated with so called turbulence closure problem. Reynolds stresses with deformation rate, and are related to viscosity, mean rate of deformation, and turbulent kinetic energy with Boussinesq’s proposal expressed can be used to calculate Reynold’s stresses in the final step of turbulence modelling. It is seen from this equation that the Reynold’s stresses are considered proportional to the dissipation rate reduced by the eddy turbulent kinetic energy. (The Kronecker delta ensures that the normal Reynolds stresses are each appropriately accounted for). It can also be seen that the kinetic energy allocates an equal third for each normal stress component (isotropic assumption). This is the reason for the inherent inaccuracy of the κ-ε model, making it incapable of describing anisotropic flow. Other scalar flow properties such as mass and heat can also be modelled using timeaveraged values. To obtain mathematical closure, the Reynolds stress terms must be related to mean flow properties either empirically or through flow model which allows calculation of this relationship. References are also sometimes made to the order of the closure. According to this terminology, a 1st order closure evaluates the Reynolds stresses through functions of the mean velocity and geometry alone, as the case above (linear). A 2 nd order closure employs a solution to a modeled form of transport partial differential equations for one or more of the characteristics of turbulence (non-linear). More complicated quadratic or cubic closures are also available through literature.
Turbulence Models Turbulence models to close the Reynolds equations can be divided originally into three groups in general. First models which (directly) use the Boussinesq assumption. Most models currently employed in engineering are this type. Experimental evidence indicates this valid in many circumstances. Second are models using the effect of closure to the Reynolds equation without this assumption. The third category is defined as those that are not based entirely on the Reynolds equation. An example would be Large Eddy Simulation (LES) where an attempt is made to resolve the
19
large scale turbulent equation from first principals be numerically solving the filtered set of equations governing this large scale. Turbulence modeling is used to approximate the effects of the sub-grid scale (SGS) turbulence. Mixing Length 0-Eqaution Baldwin-Lomax
1st - order (linear)
Reynolds Average (RANS)
1-Equation
LES
Standard
κ-ε Models
RNG Realizable
2-Equation Reynolds Stress
Turbulence Models
Spalart-Almars
2nd - order (non-linear)
Algebric Stress
DES
Quadratic & Cubic κ-ε
SST κ-ω Models Standard (Wicox)
DNS Figure 3.2
Hierarchy of Turbulence Models in General
Although such a calculation have shown great promises, it is computationally prohibited to be considered as engineer in tool11. There are other references to turbulent modeling such as order of closure. According to this terminology, a first order closure evaluates the Reynolds stresses through functions of the mean velocity and geometry alone. A second order closure employs a solution to a modeled from transport partial differential equation for one or more of the characteristics of turbulence. Since this straighter forward, we try to adapt this category as depicted Figure 3.3 Hierchey of Turbulence Models based on Relative in Figure 3.2. These are some Importance of Numerics vs. Computational Costs - (Courtsy of Tenzor) of models mentioned here and Anderson, Dale A; Tannehill, John C; Plecher Richard H; 1984,”Computational Fluid Mechanics and Heat Transfer”, Hemisphere Publishing Corporation. 11
20
by no means is it exclusive. An ideal model should be with minimum amount of complexity while capturing the essence of the relevant physics. Figure 3.3 shows the relative importance of numeric vs. computational cost for various models using a pyramid. Turbulent Viscosity, Velocity, and Length Scale Before we start the derivation for κ-ε and other models, there are some common ground that we should cover. As viscosity is the central concept for modelling the stresses, it should be noted that the turbulent viscosity μT is expressed as:
μ T ρu T l
where
uT
2 3
1 ui ui 3
Eq. 3.9
It can be seen that the turbulent viscosity is a product of density and two new variables representing turbulent velocity and turbulent length scale. Turbulent velocity uT can be described as the typical velocity occurring in the largest eddies and can also be related to the same eddies. Turbulent kinetic energy according to turbulent length scale is the average length. The new variables, uT and l form the basis for the “two-equation‟ k-ε turbulence model, meaning that in addition to the RANS equations, two more equations are required to solve for turbulent velocity and turbulent length using the model. The length scale l (for large eddies) is used in the k-ε model to define the length scale ε (for small eddies), for which a transport equation is used in the model, and represents the dissipation of the turbulent kinetic energy. The dissipation is expressed as:
ε
κ 2/3 l
as related to viscosity
ε
μ T ui u j ρ x κ x κ
Eq. 3.10
As seen in the second expression for ε, dissipation of the turbulent kinetic energy κ is proportional to the rate of deformation of eddies. Other scalar flow properties such as mass and heat can also be modelled using time-averaged values. Similar to the turbulent momentum transport’s proportionality to average velocity gradients, turbulent scalar transport is proportional to mean scalar value gradients and can be expressed as
ρu i Γ T
x i
(9.16)
Eq. 3.11
Where ГT refers to turbulent (eddy) diffusivity. As can be seen from above expression, the turbulent scalar property transport occurs with the same mechanism as in transport of momentum (mixing of eddies, represented by ГT). For this reason, it can be assumed by Reynold’s analogy that the value of ГT is similar to μT, the turbulent viscosity. The ratio of μT to ГT is defined as the Prandtl/Schmidt number σT, and has a value which is normally constant with a value around unity. The next section presents the κ-ε model and the two extra k and ε transport equations (PDEs) for closing the system of time-averaged RANS equations. The model is based on the mechanisms causing changes to turbulent kinetic energy (i.e. turbulent viscosity and velocity fluctuations). Eddy Viscosity (RANS) Models Van Driest (1956) devised a viscous damping correction for the mixing-length model. This correction
21
is still in use in most modern turbulence models. [Cebeci & Smith] refined the eddy viscosity/mixinglength concept for better use with attached boundary layers12. Algebraic Model (Zero Equation) - Mixing Length This is invariably the successful this type model which uses the Boussinesq assumption. It was originally suggested by Prandlt in 1950s a
μ T ρu T l
or
μ T ρl 2
u y
Eq. 3.12
Where l a mixing length can be thought of as a transverse distance over which the particles maintain their original momentum. For 3D thin shear layers, Prandtl’s equation is usually interpreted as: 2 2 w 2 u μ T ρl y y
1/2
Eq. 3.13
This formula treats the turbulent viscosity as a scalar and gives qualitively correct results, specially near the wall. The evaluation of l in the mixing length model varies with the type of flow being considered, wall boundary layer, jet, wakes, etc. For flow along a solid surface (internal or external), good results are observed by evaluating l according to
linner κy(1 e
y /A
) , louter C1δ
and
y
y( τ w /ρ w )1/2 νw
Eq. 3.14
Where linner predicts the inner region close to wall and Iouter exceeds linner. The κ is the von Karman constant as 0.41 and A+ is the damping constant usually set to 26. The expression for linner is reasonable for producing the inner law-of-the-wall region of turbulent flow and louter produces the outer “wake-like” region. These two zones are depicted in Figure 3.8 which shows a typical velocity distribution for an incompressible turbulent boundary layer on a smooth impermeable plate using “law-of-the-wall” coordinates. Baldwin-Lomax [Baldwin & Lomax -1978] proposed an alternative algebraic model to eliminate some of the difficulty in defining a turbulence length scale from the shear-layer thickness. It is a two-layer algebraic 0equation model which gives the eddy viscosity μt as a function of the local boundary layer velocity profile. The model is suitable for high-speed flows with thin attached boundary-layers, typically present in aerospace and turbo machinery applications. It is also commonly used in quick design iterations where robustness is more important than capturing all details of the flow physics. The Baldwin-Lomax model is not suitable for cases with large separated regions and significant curvature/rotation effects. One-Equation Model While employing a much simpler approach than two-equation or second-order closure models, oneequation models have been somewhat unpopular and have not showed a great deal of success. One notable exception was the model formulated by [Bradshaw, Ferris, and Atwell -1967], whose model 12
David C. Wilcox, “Turbulence Modeling for CFD”, 1993, 1994 by DCW Industries, Inc.
22
was tested against the best experimental data of the day at the 1968 Stanford Conference on Computation and Turbulent Boundary Layers. There has been some renewed interest in the last several years due to the ease with which one-equation models can be solved numerically, relative to more complex two-equation or second-order closure models. An obvious shortcoming of algebraic methods is that μT and uT is zero at the center of for example pipe line cases. The mixing-length model can be fixed up to overcome this using
μ T C k ρl (κ)1/2 For 2D incompressible thin - shear layer κ u C ρ(κ) Dκ μ T μ μ T D Dt y Prκ l y y 2
Eq. 3.15 3/2
Where Prκ is defined as Prandtl number for turbulence kinetic energy (≃1.0) and CD ≃ 0.164.
k - Model The complete derivation is obtained in two equation model. Spallart - Allmaras The [Spalart – Allmaras] model adds a single additional variable for a Spalart - Allmaras viscosity and does not use any wall functions; it solves the entire flow field. The model was originally developed for aerodynamics applications and is advantageous in that it solves for only a single additional variable. This makes it less memory-intensive than the other models that solve the flow field in the buffer layer. Experience shows that this model does not accurately compute fields that exhibit shear flow, separated flow, or decaying turbulence. Its advantage is that it is quite stable and shows good convergence. Two Equation Model While Kolmogorov’s κ-ω model was the first two-equation model, the most extensive work has been done by [Daly & Harlow; 1970] and [Launder & Spalding; 1972]. Launder's κ-ε model is the most widely used two-equation turbulence model; here ε is the dissipation rate of turbulent kinetic energy. Independently of Kolmogorov, [Saffman ; 1970] developed a κ-ω model that shows advantages to the more well-known κ-ε model, especially for integrating through the viscous sub-layer and in flows with adverse pressure gradients. The κ-ε model is very popular for industrial applications due to its good convergence rate and relatively low memory requirements. It does not very accurately compute flow fields that exhibit adverse pressure gradients, strong curvature to the flow, or jet flow. It does perform well for external flow problems around complex geometries. It was concluded that the algebraic and one equation models did not lend itself to application with the complex geometries that occur with internal flow calculations. In this context, we define an internal flow as a flow through turbomachinery blading, as opposed to the external flow that occurs with isolated airfoil or aircraft wing flow field predictions. When dealing with internal flows, one can overcome the limitations of a one-equation model with a two-equation model. The currently favored two-equation models are either the 𝑘-𝜀 model or the 𝑘-𝜔 model13. Both models require one to solve two transport equations to compute eddy viscosity as an algebraic expression of turbulent kinetic Alessandro Corsini, Giovanni Delibra, and Anthony G. Sheard, “A Critical Review of Computational Methods and Their Application in Industrial Fan Design”, Hindawi Publishing Corporation, ISRN Mechanical Engineering, Volume 2013, Article ID 625175. 13
23
energy (𝑘) and dissipation rate of turbulent kinetic energy (𝜀) or turbulence frequency (𝜔). In twoequation models the first equation is typically for turbulent kinetic energy (𝑘) and the second for either dissipation (𝜀) or turbulence frequency (𝜔). Both the 𝑘-𝜀 model and the 𝑘-𝜔 model rely on an assumption that one can link eddy viscosity to a time and length scale that characterizes turbulence that in turn links to the computed flow-field’s characteristics. A feature of the 𝑘-𝜀 and 𝑘-𝜔 models is that the additional transport equations for 𝑘, 𝜀, and 𝜔 share the same form, and, therefore, for a generic 𝜙 quantity, it reads
D P ε Dt x j
ν ν T σ
x j
Eq. 3.16
On the left-hand side is the quantity’s material derivative. On the right-hand side are one or more production terms, a dissipation term, a diffusion term dependent on molecular viscosity, and another given as the turbulent viscosity’s function, corrected using the Prandtl number 𝜎𝜙. The primary difference between the 𝑘-𝜀 and 𝑘-𝜔 models is the different trend of 𝜀 and 𝜔 at the wall and the definition of the wall boundary conditions for the same variables. When one studies normalized values of 𝜀 and 𝜔 for an attached flow, it is evident that 𝜔 is less dependent on the Reynolds number than 𝜀 in the wall’s near vicinity. There is a general consensus within the computational fluid dynamics community that the 𝑘-𝜀 model better reproduces the energy cascade of large-scale structures in the main flow core, whilst the 𝑘-𝜔 model performs better near the wall14. A realization that 𝑘-𝜀 models perform better in the main flow whilst the 𝑘-𝜔 models perform better near the wall leads to the natural conclusion that, ideally, one would use the two models in combination. It was observed that it is possible to combine 𝑘-𝜀 and 𝑘-𝜔 models as one can reformulate every two-equation model into every other by changing model coefficients. This realization has enabled engineers to formulate the 𝑘-𝜔 shear stress transport (𝑘-𝜔 SST) model that solves the equation for 𝜔 near the wall and 𝜀 elsewhere. The use of twoequation models has become established within the industrial community. Standard Transport Equation for k To obtain the equation for turbulent kinetic energy k, complicated algebra and rearrangements are made to the time-averaged continuity equation and the time-averaged Navier-Stokes equations for momentum. The mathematical manipulations are extensive, and therefore only a short description of what is done, followed by the resulting equations is presented. The 3 continuity equations are each multiplied by the respective fluctuating velocity component and then added together. The same process is carried out for the Reynolds equations for momentum. The two resulting equations are subtracted and rearranged extensively. Terms for viscous dissipation of turbulent kinetic energy and the Boussinesq assumption related to Reynold’s stress equation, as shown below in15.
̅̅̅̅̅̅̅̅̅̅ u′i u′j ∂(ρκ) ∂(ρu̅i κ) ∂ ∂u̅i ∂u′i ∂u′i ′ ′ ′ ′ ̅̅̅̅̅ + = + p )] − ρui uj . − μ. . [ρui . ( ⏟∂t ∂x 2 ∂xj ⏟ ∂xj′ ∂xj′ ⏟ ∂xi ⏟i ⏟ 1
Eq. 3.17
2
3
4
5
Alessandro Corsini, Giovanni Delibra, and Anthony G. Sheard, “A Critical Review of Computational Methods and Their Application in Industrial Fan Design”, Hindawi Publishing Corporation, ISRN Mechanical Engineering, Volume 2013, Article ID 625175. 15 Tennekes, H.; Lumley, J. L. “A First Course in Turbulence”, MIT Press, Cambridge, MA (1972). 14
24
Terms (3) and (5) are replaced using scalar diffusion transport terms for (3) and the time-averaged term for (5) to result in the following:
∂(ρκ) ∂(ρu̅i κ) ∂ μT ∂κ ∂u̅i ′ u′ . ̅̅̅̅̅ + = − ρ. ε [ ] − ρu ⏟ i j ⏟∂t ∂xi σκ ∂xi ∂xj ⏟ ∂xi ⏟ ⏟ 1
2
3
4
5
Eq. 3.18 Where is a constant turbulent Schmidt number for κ. The terms (1)-(5) of the turbulent kinetic energy k transport equation, can be interpreted as the following: 1. Transient term Accumulation of κ (rate of change of k). 2. Convective transport of κ by convection. 3. Diffusive transport of κ by pressure, viscous stresses, and Reynolds stresses (must be modelled). 4. Production term Rate of production of k due from the mean flow. 5. Viscous dissipation Rate of viscous dissipation of k (must be modelled) k is a constant (turbulent Schmidt number) of the k equation. The above equation for transport of κ along with the equation for transport of ε, constitute the two additional transport equations to be solved in addition to the RANS equations in the κ-ε turbulence model. The next section presents the equation for transport of ε. Standard Transport Equation for ε
∂(ρε) ∂ ∂ μT ∂ε ε ∂u̅i ε2 (ρu̅i ε) = + [ ] + C1 (−ρuu. ) − C2 ρ ⏟ ⏟∂t ∂x ∂x κ ∂xi κ ⏟i ⏟ ⏟ i σε ∂xi 1
2
3
4
5
Eq. 3.19 Similar to the transport equation for κ, the transport equation for ε includes the terms 1 - 5:
1. 2. 3. 4. 5.
Accumulation, or rate of change, of ε, Rate of destruction of ε. Diffusive transport of ε, Rate of production of ε, and Transport of ε by convection,
Cμ C1 C2 σκ σε 0.09 1.44 1.92 1.0 1.3 Table 3.1 Constant values for κε turbulence model equations
And C1, C2, σε are constants of the ε equation (see Table 3.1). Alternatively, we inscribe both equations in a more compact form, using Lagrangian Derivatives terms D/Dt:
25
Dκ Dt x j
1 μT μ σk ρ
κ μ T 2 Sij ε x j ρ
Dε ε μ (C1 T Sij2 C 2 ε) Dt κ ρ x j where μ T ρCμ
1 μT μ σε ρ
ε x j
Eq. 3.20
u j κ2 1 u and Sij i ε 2 x j x i
After obtaining κ and ε we are ready to evaluate μT as
μT
Cμ ρ(κ) 2 ε
Eq. 3.21
Despite the enthusiasm which is noted from time to time over two equation model, it is perhaps appropriate to point out two major restriction on this type of models. Since the two equation model basically turbulent viscosity models which assumes that the Boussinesq approximation holds. Therefore, its validates depends to Boussinesq approximation. In algebraic methods, μT is a local function whereas in two equation model is a more general and complex functioned governing by two additional PDEs. The second shortcoming is the need to make assumptions in evaluating the various terms in model transport equation especially third order turbulent correlations16. The same short coming plagues all other higher order closure attempts, so there is no magic bullet. The RNG Model The κ-ε model uses “highly abstruse‟ mathematics to extend the eddy viscosity turbulence model and changes the governing equations by removing smaller scales of motion, replacing them with large motions and modifying the viscosity term. Results have been good for backward-facing steps, while other results have been mixed. The RNG model is not very commonly used and has a high computational overhead. The Realizable κ-ε Model This is a non-linear version of the κ-ε model. It retains the two-equation κ-ε equations, but expands the model by including additional effects to account for Reynold’s stress anisotropy without actually using the seven extra equations used in the RSM (described later) to exactly model the Reynolds stresses. The turbulent viscosity expression and the rate of dissipation of kinetic energy equation of the standard κ-ε model are both changed in the realizable κ-ε model to take into account that turbulence does not always adjust itself instantaneously while moving through the flow domain, meaning that the Reynolds stress is partially dependent on the mean strain rate itself. This means that the non-linear realizable κ-ε model allows for the phenomena of the state of turbulence lagging behind the changes disturbing the turbulence production and dissipation balance.
Anderson, Dale A; Tannehill, John C; Plecher Richard H; 1984,”Computational Fluid Mechanics and Heat Transfer”, Hemisphere Publishing Corporation. 16
26
Issues Relating to κ-ε Turbulence Modeling Acceding to [Menter17], the κ-ε model has been very successful in a large variety of different flow situations, but it also has a number of well-known shortcomings. From the standpoint of aerodynamics, the most disturbing of them is the lack of sensitivity to adverse pressure-gradients. and separation. Under those conditions, the model predicts significantly too high shear-stress levels and thereby delays (or completely prevents) separation. This could be attributes this shortcoming to the over prediction of the turbulent length-scale in the near wall region and has shown that a correction proposed by [Hanjalic and Launder] improves the predictions considerably. However, the correction is not coordinate-invariant and can therefore not be applied in general coordinates. An alternative way of improving the results has been proposed by [Chen and Patel] and by [Rodi]. They replace the ε-equation in the near wall region by a relation that specifies the length-scale analytically. This also reduces some of the stiffness problems associated with the solution of the model. Although the procedure is coordinate independent, it has only been applied to relatively simple geometries, where the change between the algebraic relation and the e-equation could be performed along a preselected gridline. Clearly this cannot be done in flows around complex geometries. Furthermore, the switch has to be performed in the logarithmic part (the algebraic length-scale is not known in the wake region), so that the original k - ε model is still being used over most of the boundary layer. Another problem with the k- ε model is associated with the numerical stiffness of the equations when integrated through the viscous sublayer. This problem clearly depends on the specific version of the k – ε model selected, but there are some general aspects to it. All low Reynolds number k – ε models employ damping functions in one form or another in the sublayer. These are generally highly nonlinear functions k of dimensionless groups of the dependent variables like Rt= κ2/εν (models involving y+ are undesirable in separated flows). The behavior of these functions cannot easily be controlled by conventional linearization techniques and can therefore interfere with the convergence properties of the scheme. A second problem is that ε does not go to zero at a nonslip surface. There is a significant number of alternative models that have been developed to overcome the shortcomings of the κ-ε model. One of the most successful, with respect to both, accuracy and robustness, is the κω model of Wilcox18. It solves one equation for the turbulent kinetic energy k and a second equation for the specific turbulent dissipation rate (or turbulence frequency) ω. The model performs significantly better under adverse pressure-gradient conditions than the κ-ε model although it is the authors experience that an even higher sensitivity to strong adverse pressure-gradients would be desirable. Another strong-point of the model is the simplicity of its formulation in the viscous sublayer. The model does not employ damping functions and has straightforward Dirichlet boundary conditions. This leads to significant advantages in numerical stability. The κ-ω and SST Models As mentioned before, one of the main problems in turbulence modeling is the accurate prediction of flow separation from a smooth surface. Standard two-equation turbulence models, such as κ-ε models, often fail to predict the onset and the amount of flow separation under adverse pressure gradient conditions. This is an important phenomenon in many technical applications, particularly for airplane aerodynamics because the stall characteristics of a plane are controlled by the flow separation from the wing. For this reason, the aerodynamic community has developed a number of advanced turbulence models for this application. In general, turbulence models based on the ε equation predict the onset of separation too late and under-predict the amount of separation later on. This is problematic, as this behavior gives an overly optimistic performance characteristic for an Florian R. Menter, “Improved Two-Equation k-ωTurbulence Models for Aerodynamic Flows”, NASA Technical Memorandum 103975, October 1992. 18 W'llcox, D. C., "Reassessment of the Scale-Determining Equation for Advanced Turbulence Models," AIAA Journal, Vol.26, Nov. 1988, pp.1299-1310. 17
27
airfoil. The prediction is therefore not on the conservative side from an engineering stand-point. The models developed to solve this problem have shown a significantly more accurate prediction of separation in a number of test cases and in industrial applications. Separation prediction is important in many technical applications both for internal and external flows. Currently, the most prominent two-equation models in this area are the κ-ω based models of [Menter]19. The κ-ω based ShearStress-Transport (SST) model was designed to give a highly accurate predictions of the onset and the amount of flow separation under adverse pressure gradients by the inclusion of transport effects into the formulation of the eddy-viscosity. This results in a major improvement in terms of flow separation predictions. Wilcox (Standard) κ-ω Model The starting point of the present formulation is the κ-ω model developed by [Wilcox]20. It solves two transport equations, one for the turbulent kinetic energy, κ , and one for the turbulent frequency, ω. The stress tensor is computed from the eddy-viscosity concept.
Dκ ρP β*ρωκ Dt x j
ρκ κ μ σ κ ω x j
with P τ ij
ui xj
Dω γω ρκ ω ρσ d κ ω P - βρω 2 μ σ ω Dt κ x j ω x j ω x j x j
Eq. 3.22
For recommendations for the values of the different parameters, see [Wilcox ]21. The Baseline (BSL) κ-ω Model The main problem with the Wilcox model is its well-known strong sensitivity to freestream conditions [Menter]22. Depending on the value specified for ω at the inlet, a significant variation in the results of the model can be obtained. This is undesirable and in order to solve the problem, a blending between the κ-ω model near the surface and the κ-ε model in the outer region was developed by [Menter]23. It consists of a transformation of the κ-ε model to a κ-ω formulation and a subsequent addition of the corresponding equations. The Wilcox model is thereby multiplied by a blending function F1 and the transformed κ-ε model by a function (1 - F1), F1 is equal to one near the surface and decreases to a value of zero outside the boundary layer (that is, a function of the wall distance). At the boundary layer edge and outside the boundary layer, the standard κ-ε model is therefore recovered24. SST κ-ω Model The Shear-Stress Transport (SST) κ-ω turbulence model is a type of hybrid model, combining two models in order to better calculate flow in the near-wall region. It was designed in response to the problem of the κ-ε models unsatisfactory near-wall performance for boundary layers with adverse Menter, F.R., “Two-equation eddy-viscosity turbulence models for engineering applications”, AIAA-Journal., 32(8), pp. 1598 - 1605, 1994. 20 Wilcox, D. C. (2008), “Formulation of the k–ω Turbulence Model Revisited”, 46 (11), AIAA Journal, pp. 2823– 2838, Bibcode:2008AIAAJ..46.2823W, doi:10.2514/1.36541. 21 Wikipedia. 22 Menter, F.R., “Multiscale model for turbulent flows”, In 24th Fluid Dynamics Conference. American Institute of Aeronautics and Astronautics, 1993. 23 Menter, F.R., “Two-equation eddy-viscosity turbulence models for engineering applications”, AIAA-Journal., 32(8), pp. 1598 - 1605, 1994. 24 Training Manual, ANSYS CFX. 19
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pressure gradients. It utilizes a standard κ-ε model to calculate flow properties in the free-stream (turbulent) flow region far from the wall, while using a modified k-ε model near the wall using the turbulence frequency ω as a second variable instead of turbulent kinetic energy dissipation term ε. Since the air in this projects heat exchanger is flowing between two flat fins very close to each other, it is expected that the boundary layer flow has a strong influence on the results, and properly modelling this near-wall flow could be important for accuracy of the calculations. Therefore the SST κ-ω turbulence model has also been chosen for CFD simulations in this project. This SST κ-ω model is similar to the κ-ε turbulence model, but instead of ε as the second variable, it uses a turbulence frequency variable omega, which is expressed as ω = ε/k [s-1]. The SST κ-ω model computes Reynolds stresses in the same way as in the κ-ε model. The transport equation for turbulent kinetic energy k for the k-ω model is:
∂(ρκ) ∂ ∂ μT (ρu̅i κ) = + [(μ + ) ∇κ ] + P⏟κ − β⏟∗ ρκω ⏟∂t ∂x ∂x σκ ⏟i ⏟i 1
2
4
3
where Pκ = (2μT
5
∂u̅i ∂u̅i 2 ∂u̅i . − ρκ δ ) ∂xj ∂xj 3 ∂xj ij
Eq. 3.23 The terms (1) - (5) in above expression, the turbulent kinetic energy k transport equation for the SST Omega turbulence model, can be interpreted as the following: 1. Transient term - Accumulation of k (rate of change of k) 2. Convective transport - Transport of k by convection 3. Diffusive transport - Turbulent diffusion transport of k 4. Production term - Rate of production of k 5. Dissipation Rate of dissipation of k Where σκ and β* are equation constants. The transport equation for turbulent frequency ω for the kω model is:
∂(ρω) ∂(ρu̅i ω) ∂ μT + = [(μ + ) ∇ω] + ⏟∂t ∂x σω,1 ⏟ ∂xi ⏟i 1
2
3
∂u̅i ∂u̅i 2 ∂u̅i ρ ∂κ ∂ω γ2 (2ρ . − ρω δij ) − β⏟2 ρω2 + 2 ∂xj ∂xj 3 ∂xj σ ⏟ω,2 ∂xκ ∂xκ ⏟ 4
5
6
Eq. 3.24 The general description for each of the terms in (1) to (6) are the usual terms for accumulation, convection, diffusion, production, and dissipation of ω. The last term (6) is called a “cross-diffusion‟ term, an additional source term, and has a role in the transition of the modelling from ε to ω. The constants for the Mentor SST κ-ω turbulence model are listed in β* β2 σκ σω,1 σω,2 Υ2 Table 3.2. Additional modifications have been made to 0.09 0.083 1.0 2.0 1.17 0.44 the model for performance optimization. There are blending functions added to improve the numerical Table 3.2 Constant values for SST κ-ω turbulence model equations stability and make a smoother transition between the two models. There have also been limiting functions
29
made to control the eddy viscosity in wake region and adverse pressure flows25. The κ-ω model is similar to κ-ε, but it solves for ω, the specific rate of dissipation of kinetic energy. It also uses wall functions and therefore has comparable memory requirements. Additionally, it has more difficulty converging and is quite sensitive to the initial guess at the solution. Hence, the κ-ε model is often used first to find an initial condition for solving the κ-ω model. The κ-ω model is useful in many cases where the κ-ε model is not accurate, such as internal flows, flows that exhibit strong curvature, separated flows, and jets26. Case Study – Subsonic Flow for Turbulence Models of (NACA)-0012 Airfoil The analysis of the 2D subsonic flow over a (NACA) 0012 airfoil at various angles of attack and operating at a Re = 3×106 is presented by [Eleni et al.]27. The flow was obtained by solving the steadystate governing equations combined with one of three turbulence models (Spalart-Allmaras, Realizable κ-ε, and κ-ω Shear Stress Transport (SST). The aim of the work was to show the behavior of the airfoil at these conditions and to establish a verified solution method. The computational domain was composed of 80000 cells emerged in a structured way, taking care of the refinement of the grid near the airfoil in order to enclose the boundary layer approach. Calculations were done for constant air velocity altering only the angle of attack for every turbulence model tested. This work highlighted two areas in (CFD) that require further investigation: transition point prediction and turbulence modeling. The laminar to turbulent transition point was modeled in order to get accurate results for the drag coefficient at various Reynolds numbers. In addition, calculations showed that the turbulence models used in commercial CFD codes does not give yet accurate results at high angles of attack. Computational Method The NACA 0012, with Reynolds number of Re = 3x106, same with the reliable experimental data from [Abbott and Von Doenhoff]28, in order to validate the present simulation. The free stream temperature is 300 K, which is the same as the environmental temperature. The density of the air at the given temperature is ρ=1.225kg/m3 and the viscosity is μ=1.7894×10-5 kg/ms. For this Reynolds number, the flow can be described as incompressible. This is an assumption close to reality and it is not necessary to resolve the energy Figure 3.4 Lift Coefficient at Stall (AoA) against Number of Cells equation. A segregated, Versteeg, H K; Malalasekera, W.”An Introduction to Computational Fluid Dynamics, The Finite Volume Method”, Second edition, Pearson Education Limited, Essex, England (2007). 26 Walter Frei, “Which Turbulence Model Should I Choose for My CFD Application? “, COMSOL Blog, September 16, 2013. 27 D. C. Eleni, Tsavalos I. Athanasios and Margaris P. Dionissios, “Evaluation of the turbulence models for the simulation of the flow over a National Advisory Committee for Aeronautics (NACA) 0012 airfoil”, Journal of Mechanical Engineering Research, March 2012. 28 Abbott IH, Von Doenhoff AE. “Theory of Wing Sections”. Dover Publishing, New York. 25
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implicit solver was utilized where the calculations were done for angles of attack ranging from -12 to 20. The first step in performing a CFD simulation should be to investigate the effect of the mesh size on the solution results. The appropriate number of nodes can be determined by increasing the number of nodes until the mesh is sufficiently fine so that further refinement does not change the results (mesh independence). Figure 3.4 shows the effect of number of grid cells in coefficient of lift at stall angle of attack (16°). This study revealed that a C-type grid topology with 80000 quadrilateral cells would be sufficient to establish a grid independent solution. The domain height was set to approximately 20 chord lengths, and the height of the first cell adjacent to the surface was set to 105, corresponding to a maximum y+ of approximately 0.2. A y+ of this size should be sufficient to properly resolve the inner parts of the boundary layer. In order to include the transition effects in the aerodynamic coefficients calculation and get accurate results for the drag coefficient, a new method was used. The transition point from laminar to turbulent flow on the airfoil was determined and the computational mesh was split in two regions, a laminar and a turbulent region. To calculate the transition point the following procedure was used. A random value for the transition point (xtr) was chosen and the computational domain was split at that point with a perpendicular line. The problem was simulated by defining the left region as laminar and the right as turbulent zone. Results and Discussion Simulations for various angles of attack were done in order to be able to compare the results from the different turbulence models and then validate them with existing experimental data from reliable sources. To do so, the model was solved with a range of different angles of attack from -12 to 20°. On an airfoil, the resultants of the forces are usually resolved into two forces and one moment. The component of the net force acting normal to the incoming flow stream is known as the lift force and the component of the net force acting parallel to the incoming flow stream is known as the drag force.
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The curves of the lift and the drag coefficient are shown for various angles of attack, computed with three turbulence models and compared with experimental data. Figure 3.5 shows that at low angles of attack, the dimensionless lift coefficient increased linearly with angle of attack. Flow was attached to the airfoil throughout this regime. At an angle of attack of roughly 15 to 16°, the flow on the upper surface of the airfoil began to separate and a condition known as stall began to develop. All three models had a good agreement with the experimental data at angles of attack from -10 to 10° and the same behavior at all angles of attack until stall. It was obvious that the Spalart-Allmaras turbulence model had the same behavior with the experimental data as well as after stall angle.
Figure 3.5 Comparison Between Experimental Data [Abbott et al.] and Three Different Turbulent Models Simulation Results for Lift Coefficient in NACA 0012 Airfoil
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Near stall, disagreement between the data was shown. The lift coefficient peaked and the drag coefficient increased as stall increased. The predicted drag coefficients were higher than the experimental data (Figure 3.6). This over prediction of drag was expected since the actual airfoil has laminar flow over the forward half. The turbulence models cannot calculate the transition point from laminar to turbulent and consider that the boundary layer is turbulent throughout its length. From theory, the turbulent boundary layer carries more energy and is much greater than at the viscous boundary layer, which carries less energy. The computational results must be compared with experimental data of a fully turbulent boundary layer. This was done only for CD as CL is less sensitive to the transition point. [ Johansen]29 contained experimental data of CD for the NACA 0012 airfoil and Re = 3×106, where the boundary layer formed around the airfoil is fully turbulent. Figure 3.6 shows the curves of CD for various angles of attack, compared with experimental data for fully turbulent boundary layer30. The values of from the three turbulence models were very close to experimental data for the fully turbulent boundary layer. The most accurate model was the κ-ω SST model, next came the SpalartAllmaras, and latest in precision was the Realizable κ-ε . In order to get more accurate results, the computational domain could be split into two different domains to run mixed laminar and turbulent flow. The disadvantages of this approach were that the accuracy of simulations depends on the ability to accurately guess the transition location, and a new grid must be generated if the transition point had to change [Silisteanu-Botez]31. If the transition point is known, the grid can easily be split in two with a vertical line that passes through this point and then laminar and turbulent zones are defined. The results of this method at angle of attack a=0 and operating at Re = 1×106, 2×106, 3×106, 4×106 and 5×106. Initially, was calculated for a fully turbulent boundary layer and compared with CD experimental data from NASA [McCroskey]32. Then, simulations were made with the split grid for the five Reynolds numbers. The computational results for the fully turbulent boundary layer agreed very well with the corresponding experimental data. The discrepancy between the Drag Coefficient and experimental data from [McCroskey] for fully turbulent boundary layer was up to 5.6%. On the other hand, the comparison between the simulation Figure 3.6 Comparison Between Experimental Data for results with the split grid and the Transitional Boundary Layer and Different Turbulent Models experimental data from [McCroskey] on the Drag Coefficient of NACA-0012 Airfoil Johansen J. “Prediction of Laminar/Turbulent Transition in Airfoil Flows”. RISE National Laboratory, Roskilde, Denmark, 1997. 30 Comparison Between Different Turbulent Models and Experimental Data obtained by [Abbott & Von Doenhoff] and [Johansen] for Transitional Boundary Layer on the Drag Coefficient of NACA-0012 Airfoil. 31 Silisteanu PD, Botez RM. “Transition flow occurrence estimation new method”. 48th AIAA Aerospace Science Meeting. Orlando, Florida, 2010. 32 McCroskey WJ, ”A Critical Assessment of Wind Tunnel Results for the NACA 0012 Airfoil”. U.S. Army Aviation Research and Technology Activity, Nasa Technical Memorandum, 42: 285-330, 1987. 29
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for transitional boundary layer showed an excellent agreement, with maximum error of about 3.6%. It was also observed that as the Reynolds number increased, the Drag Coefficient decreased. When the boundary layer was fully turbulent the reduction of was more intense and when there was a transition from laminar to turbulent, was reduced to a much lower rate. It is worth noting that this process of calculating the transition point is quite simple when the angle of attack is zero because the flow is symmetric and the transition point is the same above and below the airfoil. At nonzero angles of attack the process is more complicated because transition points are different for the upper and lower surface of the airfoil. With different AoA, the pressure on the lower surface of the airfoil was greater than that of the incoming flow stream and as a result it effectively “pushed” the airfoil upward, normal to the incoming flow stream. On the other hand, the components of the pressure Figure 3.7 Contours of velocity magnitude at 9° (Top) and 16° distribution parallel to the incoming (Bottom) AoA with the Spalart-Allmaras turbulence model flow stream tended to slow the velocity of the incoming flow relative to the airfoil, as do the viscous stresses. Contours of velocity components at angles of attack 9 and 16° are also shown in Figure 3.7. The trailing edge stagnation point moved slightly forward on the airfoil at low angles of attack and it jumped rapidly to leading edge at stall angle. A stagnation point is a point in a flow field where the local velocity of the fluid is zero. The upper surface of the airfoil experienced a higher velocity compared to the lower surface. That was expected from the pressure distribution. As the angle of attack increased the upper surface velocity was much higher than the velocity of the lower surface. For further information, please consult the [Eleni et al.]33. Conclusions This paper showed the behavior of the 4-digit symmetric airfoil NACA 0012 at various angles of attack. The most appropriate turbulence model for these simulations was the κ-ω SST two-equation model, which had a good agreement with the published experimental data of other investigators for a wider range of angles of attack. The predicted drag coefficients were higher than the existing experimental data from reliable sources. This over prediction of drag was expected since the actual D. C. Eleni, Tsavalos I. Athanasios and Margaris P. Dionissios, “Evaluation of the turbulence models for the simulation of the flow over a National Advisory Committee for Aeronautics (NACA) 0012 airfoil”, Journal of Mechanical Engineering Research, March 2012. 33
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airfoil has laminar flow over the forward half. The computational results from the three turbulence models were compared with experimental data where the boundary layer formed around the airfoil is fully turbulent and they agreed well. Afterwards, the transition point from laminar to turbulent regime was predicted, the computational grid split in two regions, a laminar and a turbulent region, and then new simulations were realized. By this method, the computational results agreed very well with corresponding experimental data34. Near Wall Turbulence Modeling Most turbulence models requires special algebraic formula, often called Wall Function to represent the distribution of velocity, temperature, turbulence, energy, etc. within boundary layers. This practice is necessary because these models are not valid in the region within the layer where the molecular and turbulent effects are comparable in magnitude. It is also expedient because it avoids the need for employing a fine mesh within boundary layer. The wall function representation of nearwall turbulent behavior is inexact, the accuracy being dependent on the degree to which the assumption and approximations embodied in the function correspondent with reality of the application. The Standard wall function with following characteristics:
Variation in velocity etc. are predominantly normal to the wall, leading to one dimensional
Figure 3.8
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Zones in Turbulent B. L. for a typical Incompressible flow over a smooth flat plate
behavior. Effects of pressure gradients and body forces are negligible, leading to uniform shear stress in the layer. Shear stress and velocity vectors are aligned and unidirectional through the layer. A balance exist between turbulence energy production and dissipation.
See Previous.
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There is a linear variation of turbulence length scales.
For flow along solid surfaces (internal or external), a good approximation is observed by evaluating the expressions for linner is responsible for producing the inner law of the wall region of turbulent flow and louter produces the outer wake-like region. These two zones are illustrated in Figure 3.8 which depicts a typical velocity distribution for an incompressible turbulent boundary layer on a smooth impermeable plate. Modeling Flow Near the Wall Two approaches are commonly used to model the flow in the near-wall region: 1 - Wall Function Method uses empirical formulas that impose suitable conditions near the wall without resolving the boundary layer, thus saving computational resources. All turbulence models address a suitable wall function method. The major advantages of the wall function approach is that the high gradient shear layers near walls can be modeled with relatively coarse meshes, yielding substantial savings in CPU time and storage. It also avoids the need to account for viscous effects in the turbulence model. (see Figure 3.9). 2 - Low-Reynolds-Number Method Resolves the details of the boundary layer profile by using very small mesh length scales in the direction normal to the wall (very thin inflation layers). Note that the low-Re method does not refer to the device Reynolds number, but to the turbulent Reynolds number, which is low in the viscous sublayer. This method can therefore be used even in simulations with very high device Reynolds numbers, as long as the viscous sublayer has been resolved. The computations are extended through the viscosity-affected sublayer close to the wall. The low-Re approach requires a very fine mesh in the near-wall zone and correspondingly large number of nodes. Computer-storage and run-time requirements are higher than those of the wall-function approach and care must be taken to ensure good numerical resolution in the near-wall region to capture the rapid variation in variables. To reduce the resolution requirements, an automatic wall treatment was developed which allows a gradual switch between wall functions and low-Reynolds number grids, without a loss in accuracy.
Figure 3.9
Turbulence and near Wall Function
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Case Study - Wall-modelling strategies in Large Eddy Simulation of separated highReynolds-number flows If ReL (say, that based on the boundary layer thickness) exceeds roughly 105, the resource requirements are entirely dominated by the need to resolve the near-wall region (the inner layer), with the number of nodes rising roughly as N ~ Re2.5L. Simulations for practical configurations would not, in most circumstances, be undertaken today with meshes exceeding 10–50M nodes, corresponding to ReL ~ 5 x 105, yet this is still a very modest Reynolds number in practice [Tessicini, and Leschziner]35. The general approach taken in recent years towards addressing the problem of near-wall resolution has been to combine RANS modelling near the wall with LES in the outer flow. The key premise underpinning this strategy is that it should allow near-wall numerical cells to be used that have far higher aspect ratios than those required by LES, typically 500-1000, relative to 50 in wall-resolving LES. Substantial savings could thus be made by using much coarser stream wise and span wise meshes than are dictated by the LES constraints. Efforts currently have focused on two particular methods, one being a hybrid LES-RANS scheme and the other being a two-layer zonal schemes. The difference between them is explained by reference to Figure 3.10, which conveys the manner by which the LES and RANS regions communicate numerically. The hybrid method uses a single computational domain. Within a predefined layer near the wall, that can be prescribed in terms of y+, RANS equations are solved using one-equation or two-equation eddy-viscosity models that are dynamically adjusted so to comply with continuity of eddy viscosity across the interface, νRANS= νLES, beyond which a LES sub-grid-scale model is used. To achieve this compatibility, the RANS model coefficients at the interface is determined by comparison of the RANS viscosity, containing the RANS-determined turbulence energy and dissipation rate at the interface, to the LES viscosity at the interface. The variation of the coefficients from the interface to the wall is then prescribed analytically, based on observations derived from a-priori wall-resolved LES
Figure 3.10
Schematics of hybrid LES-RANS scheme (upper) and two-layer zonal scheme (lower)
F. Tessicini, M.A. Leschziner, “Wall-modelling strategies in large eddy simulation of separated high- Reynoldsnumber flows”. 35
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performed in channel flows. The zonal method uses two overlapping grids across the near-wall layer. The LES grid extends to the wall, but is relatively coarse, maintaining cell-aspect-ratio constraints appropriate to LES. Within the near-wall layer, a separate grid is inserted, which is refined towards the wall, typically to a wall-nearest node located at y+=O(1). Within that layer, parabolized RANS equations are solved for the wall-parallel-velocity components, using a simple algebraic turbulence model - for example, a mixing-length model.
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4 Beyond the Boussinesq Approximation (Non-Linear Modeling) The Boussinesq eddy-viscosity approximation assumes the principal axes of the Reynolds-stress tensor, τij, are coincident with those of the mean strain-rate tensor, Sij, at all points in a turbulent flow. This is the analog of Stokes approximation for laminar flows. The coefficient of proportionality between τij and Sij is the eddy viscosity, μT, which is linear. Unlike the molecular viscosity which is a property of the fluid, the eddy viscosity depends upon many details of the flow under consideration. It is affected by the shape and nature (e.g., roughness height) of any solid boundaries, freestream turbulence intensity, and, perhaps most significantly, flow history effects. Experimental evidence indicates that flow history effects on τij often persist for long distances in a turbulent flow, thus casting doubt on the validity of a simple linear relationship between τij and Sij. Next, we outline several flows for which the Boussinesq approximation yields a completely unsatisfactory description.
Boussinesq-Approximation Deficiencies
While models based on the Boussinesq eddy-viscosity approximation provide excellent predictions for many flows of engineering interest, there are some applications for which predicted flow properties differ greatly from corresponding measurements. Generally speaking, such models are inaccurate for flows with sudden changes in mean strain rate and for flows with what Bradshaw refers to as extra rates of strain. It is unsurprising that flows with sudden changes in mean strain rate pose a problem. The Reynolds stresses adjust to such changes at a rate unrelated to mean flow processes and time scales, so that the Boussinesq approximation must fail. Similarly, when a flow experiences extra rates of strain caused by rapid dilatation, out of plane straining, or significant streamline curvature, all of which give rise to unequal normal Reynolds stresses, the approximation again becomes suspect. Some of the most noteworthy types of applications for which models based on the Boussinesq approximation fail are: 1. 2. 3. 4. 5. 6.
flows with sudden changes in mean strain rate; flow over curved surfaces; flow in ducts with secondary motions; flow in rotating and stratified fluids ; three-dimensional flows; flows with boundary-layer separation .
Nonlinear Constitutive Relations
One approach to achieving a more appropriate description of the Reynold stress tensor without introducing any additional differential equations is to assume the Boussinesq approximation is simply the leading term in a series expansion of functional. Proceeding with this premise it can be shown that for incompressible flow the expansion must proceed through second order according to
ui u j 2 δ ij ρ κ ρu i uj μ T x 3 x i j 2 ρu i uj TSij δ ij κ 3
where
or
u j 1 u Sij i 2 x j x i
Eq. 4.1
The above isotropic relation assumes that the principal axis of the Reynolds stress tensor S͞ ij coincides with that of the mean strain rate. The standard κ-ε model does not take into account the anisotropic effects and fails to represent the complex interaction mechanisms between Reynolds stresses and
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the mean velocity field. For example, the linear model fails to mimic the effects related to streamline curvature, secondary motion, or flow with extra strain rates. These anisotropic effects can be predicted by introducing a nonlinear expression for the Reynolds stresses as given in the following expression36:
2 ρ̅κ ρ̅κ ′ u′ = − ρ ̅̅̅̅̅̅̅ −ρu ̅ κδij + 2μT S̅ij − B 2 S̅mn S̅nm δij − C 2 Sik Skj − i j ⏟3 ⏟ω ω Boussinesq
Non−Linear Trem 1
ρ̅κ ρ̅κ ρ̅κ ̅ kj + S̅jk Ω ̅ ki ) − F ̅ mn Ω ̅ nm δij − G ̅ Ω ̅ D 2 (S̅ik Ω Ω Ω ⏟ω ω2 ω2 ik kj Non−Linear Term 2
∂u̅ κ κ3 1 ∂u̅ ̅ ij = ( i + j ) where ≈ , S ω 2 ε2 2 ∂xj ∂xi
∂u̅ 1 ∂u̅ ̅ ij = ( i − j ) and Ω 2 ∂xj ∂xi
Eq. 4.2 Where B, C, D, F, and G are closure coefficients. These coefficients of the non-linear terms should be carefully determined because they are expected to influence the physical accuracy and numerical performance of the model. Here, the coefficients are adjusted through the consideration of the anisotropy in simple shear flows detailed by [Champagne et al.]37 and [Harris et al.]38 Solving this equation is a daunting task, but there are assumptions (as the case with all turbulence models) which can be made, depending to the case. We don’t get into different modeling but for an excellent discussion, readers should refer to39. (See Eq. 4.2). Case Study – 3D Simulation of Flow Past a Cylinder using Nonlinear Turbulence Model The flow past a cylinder of circular cross section has been the subject of interest for industrial researchers as well as scientists, because of its wide range of applications. To cite a few examples, flow in bridge piers, chimney stacks, and tower structures in civil engineering; electrodes in chemical engineering; nuclear fuel rods in the atomic field and heat exchanger tubes in thermal engineering, etc., fall under this subject of study. Although the geometry is simple, the flow has complicated features such as stagnation points, laminar boundary-layer separation, turbulent shear layers, periodic vortex shedding, and wakes. Even though there is much literature available on numerical simulation of laminar flow past a two-dimensional circular cylinder at low Reynolds number, a focus on practical high Reynolds numbers is less. This could be due to the complexity of formulating Reynolds stresses in turbulent flows. The majority of turbulent flow calculations carried out in earlier days used two equation models such as the standard κ-ε model (hereafter referred as SKE) and the κ-ω model, because of their robustness, computational efficiency, and completeness. In the classical SKE model, the turbulent kinetic energy (κ) and the turbulent kinetic energy dissipation rate (ε) were calculated using modeled transport equations separately for k and e along with the Boussinesq eddy viscosity approximation.
Ichiro Kimura, and Takashi Hosoda, ”A non-linear κ-ε model with realizability for prediction of flows around bluff bodies”, Int. J. Numerical Meth. Fluids 2003; 42:813–837 (DOI: 10.1002/_d.540). 37 Champagne FH, Harris VG, Corrsin S. ,”Experiments on nearly homogeneous turbulent shear flow”. Journal of Fluid Mechanics 1970; 41:81–139. 38 Harris VG, Graham JAH, Corrsin S. “Further experiments in nearly homogeneous turbulent shear flow”. Journal of Fluid Mechanics 1977; 81:657–687. 39 David C. Wilcox, ”Turbulence Modeling for CFD”,1993, 1994 by DCW Industries, Inc. 36
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Literature Survey Subsequently, nonlinear models were proposed by many researchers such as [Gatski and Speziale]40, [Craft et al].41, and Shih et al42. Some researchers have tested the appropriateness of nonlinear turbulence models for simple flows such as homogeneous shear flow, separated flow over a backward-facing step, and flow in a confined jet. They obtained encouraging results in the prediction of mean and turbulent quantities. The two-dimensional, unsteady, Reynolds-averaged Navier-Stokes (URANS) simulation of subcritical flow past a circular cylinder at Re = 1.4x105 was carried out by [Tutar and Holdo]43 using the SKE, nonlinear κ-ε formulation and large-eddy simulation (LES) technique. The wake centerline velocity recovery predicted by the nonlinear κ-ε turbulence model showed better agreement with the LES and experimental results than those predicted by the SKE model. [Saghafian et al.]44 simulated the two dimensional flow past a circular cylinder using a nonlinear eddy viscosity model and tested the mean drag coefficient (Cd mean), root-mean-square (RMS) lift coefficient (Cl RMS), and Strouhal number value (St = f D/U∞) for the range of Reynolds numbers from subcritical laminar separation to supercritical turbulent separation. The standard and renormalized group versions of the κ-ε model were examined by Jennifer 45 for two-dimensional flow past a circular cylinder at subcritical Reynolds number Re=5,232. With this increasing interest in nonlinear turbulence models, [Kimura and Hosoda]46 proposed a cubic nonlinear κ-ε model by accounting for the effect of anisotropy. They tested the model for twodimensional flow around a square cylinder and a surface-mounted cubic obstacle. The model, which included the realizability condition, performed better than linear models when compared with experimental results. Recently, the above model was used by [Ramesh et al.]47 for simulating three dimensional flow around a square cylinder, and they found it performs better than results by the standard κ-ε and RNG κ-ε models. Basic Equations The ensemble-averaged RANS equations for an incompressible flow are Continuity and Momentum equations given by:
u i 0 x i
,
u i u i u j P (ui uj ) 2ui Re 1 t i x j x i x j x i x j
Eq. 4.3
where xi is the spatial coordinate, t is the time, u͞i is the ensemble-averaged velocity, ͞u´i is the 40 T. B. Gatski and C. G. Speziale,
“On Explicit Algebraic Stress Models for Complex Turbulent Flows”, J. Fluid Mech., vol. 254, pp. 59–78, 1993. 41 T. J. Craft, B. E. Launder, and K. Suga, “Development and Applications of a Cubic Eddy-Viscosity Model of Turbulence”, Int. J. Heat Fluid flow, vol. 17, pp. 108–115, 1996. 42 T. H. Shih, J. Zhu, and J. L. Lumley, “A New Reynolds Stress Algebraic Equation Model”, Computer. Meth. Appl. Mech. Eng., vol. 125, pp. 287–302, 1997. 43 M. Tutar and A. E. Holdo, “Computational Modeling of Flow around a Circular Cylinder in Subcritical Flow Regime with Various Turbulence Models”, Int. J. Numerical Methods Fluids, vol. 35, pp. 763–784, 2000. 44 M. Saghafian, P. K. Stansby, M. S. Saidi, and D. D. Asplay, “Simulation of Turbulent Flows around a Circular Cylinder Using Non-Linear Eddy-Viscosity Modeling: Steady and Oscillatory Ambient Flows”, J. Fluids Structure, vol. 17, pp. 1213–1236, 2003. 45 R. B. Jennifer, “Verification Testing in Computational Fluid Dynamics: An Example Using Reynolds-Averaged Navier-Stokes Methods for Two Dimensional Flow in the Near Wake of a Circular Cylinder”, Int. J. Numerical Meth. Fluids, vol. 43, pp. 1371–1389, 2003. 46 L. Kimura and T. Hosoda, “A Non-linear κ-ε Model with Reliability for Prediction of Flows around Bluff Bodies”, Int. J. Numerical Method Fluids, vol. 42, pp. 817–837, 2003. 47 V. Ramesh, S. Vengadesan, and J. L. Narasimhan, “3D Unsteady RANS Simulation of Turbulent Flow over Bluff Body by Non-linear Model”, Int. J. Numerical Meth. Heat Fluid Flow, vol. 16, pp. 660–673, 2006.
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fluctuating velocity, and P is the averaged pressure divided by the density. As a result of ensembleaveraging process, further unknowns are introduced into the momentum equations by means of Reynolds stresses ͞u´iu´j . In engineering flows, closure approximation using two-equation models for u ͞ ´ju´j have gained popularity because of their simplicity. In this article the study is confined to the κ-ε model, which employs additional transport equations for turbulent kinetic energy κ and its dissipation rate ε, and they are given as
κ κu j u ui uj i ε t x j x j x j
ν t κ ν σ κ x j
ε εu j ε ε2 C1 u i u j C 2 t x j κ κ x j
ν t ε ν σ ε x j
Eq. 4.4
Where κ is the turbulent kinetic energy, is the turbulent kinetic energy dissipation rate, ν is the fluid kinematic viscosity, and νt is the eddy viscosity. C1, C2, σκ, and σε are the model constants given by Table 3.1. Numerical Strategies The governing equations for velocities and turbulent quantities are solved using the finite-volumebased commercial solver FLUENT® 6.2. The equations are discretized on a collocated grid in fully implicit form. Momentum equations are solved using the QUICK scheme, and the SIMPLE algorithm is used for coupling the pressure and velocity terms. The second-order upwind scheme is used to discretize convective terms and also the terms in equations for turbulent quantities. The secondorder implicit scheme is used for time integration of each equation. The present nonlinear model is incorporated in FLUENT through User-Defined Functions (UDFs). The nonlinear stress term is added as a source term in equations for κ and ε. The turbulent viscosity is also made to vary. The implementation of the present Nonlinear κ-ε model (hereafter referred to as NLKE) was validated for turbulent flow around a square cylinder. A boxtype computational domain (Figure 4.1) with structured grid in Cartesian coordinates having origin at the center of the cylinder is used. Stream wise direction is along the x axis with x ¼ 0 at the center of the cylinder, the y axis is the vertical axis with y ¼ 0 the wake centerline, and the z axis is the span wise direction with z ¼ 0 being the mid span of the cylinder. The upstream boundary with uniform inlet velocity is placed at 7D from the center of the body, where D is the diameter of the cylinder. This extent is less than that used in the LES simulation of FF02, where the inlet boundary was kept at 10D from center of the body and the URANS, and the LES simulation of LU01, who used 8D from the center of the body. At the inlet boundary, 2% turbulence intensity is specified. The convective boundary condition at the outlet is specified at 20D downstream of the body. The same domain size was used by FF02, whereas LU01 used 24D. In the y direction, slip boundary condition is applied at 8D from the center of the body, which is the same as the one used by previous references. Most other reported LES simulations have used either an O-grid or a C-grid with larger boundary dimensions. All the LES simulations were done with span wise domain size of PD, whereas the URANS simulations have been reported only for two-dimensional flow. The three-dimensional stability analysis and the experiment on a subcritical Reynolds number showed the dominant span wise scales having wavelengths of approximately three to four cylinder diameters in the Reynolds number range 180 < Re < 240. After this Reynolds number, the wavelength shortens to nearly one diameter. In our simulation, we have taken a span wise length of 4D along the z direction, and periodic boundary condition is enforced on the boundary. This extent is slightly larger than that used in LES and DNS calculations. A nonequilibrium wall function approach is used to capture the adverse pressure gradient effect.
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Geometry and Meshing The present NLKE model was simulated with five different grids for grid dependence studies. In all grids, along the circumference of the cylinder, 144 mesh points were placed, and in the cross-stream direction, 85 mesh points were used. For all grids, the mesh points were placed uniformly in the span wise direction. Different grids were achieved by systematically changing the resolution in the stream wise and span wise directions. Figure 4.1 shows the schematic representation of the computational domain used. Due to ease of domain, a simple structured grid being used. Of course, to be noted that the overall grid size is coarser than those used by LES and DNS. The same grid is used to simulate three-dimensional flow by the standard κ-ε model (referred to as 3DSKE) as well as twodimensional flow by the nonlinear κ-ε model (2DNLKE) for comparison purposes. Results and Discussion The solution is initiated and allowed to march in time with a non-dimensional increment of dt (Δt U∞/D =5x103) until the vortex shedding becomes periodic. The time variation of the lift coefficient is shown in Figure 4.4, and periodicity is observed. The simulation is continued for five more vortex shedding cycles to advect all the numerical errors to Figure 4.1 Schematic Representation of the Computational Domain downstream. In Breuer’s LES work 48, in the time history of the lift coefficient, there was a low-frequency component over a regular periodic component. Hence, in order to achieve reproducible statistics, averaging was done over 22 vortex shedding cycles. In the present work, as there are no such features, the time averaging is done over 10 vortex shedding cycles to obtain both bulk and mean field quantities. Here T (t U∞/D) is non dimensional time for one vortex shedding cycle. The present three-dimensional simulation with the NLKE model (referred as 3DNLKE) shows better agreement with the experimental results. The present 3DNLKE simulation predicted the mean drag coefficient (Cd mean) and Strouhal number (St) within the experimental uncertainty, but the two-dimensional simulation (2DNLKE) under predicts these values. The Figure 4.2 Variation of lift Coefficient with Time M. Breuer, “Large Eddy Simulation of the Subcritical Flow past a Circular Cylinder: Numerical and Modeling Aspects”, Int. J. Numerical Meth. Fluids, vol. 28, pp. 1281–1302, 1998. 48
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recirculation length (Lr= D) and negative velocity (U1 min= U∞) in the wake predicted by the present 3DNLKE simulation agreed well with those of experimental and LES results. Simulation by 3DSKE predicts longer recirculation length (Lr=D) and less negative velocity in the wake. Location of the minimum velocity (rmin=D) by the present two-dimensional simulation with the NLKE model (2DNLKE) and the two-dimensional URANS simulation result from LU01 showed an under predicting trend. However, the present 3DNLKE results match well with experiment. The mean stream wise velocity recovery along the wake centerline is shown in Figure 4.3. It can also be observed that the experimental data showed some scatter. Except for these locations, the present 3DNLKE results are in good agreement Figure 4.3 Comparison of Wall Pressure Coefficient (Cp) with experiment. However, slower wake velocity recovery is observed with 3DSKE model. at this Reynolds number and beyond ineffective, as variations in the span wise direction are neglected. Figure 4.2 provides a comparison of the wall pressure coefficient with experimental and different numerical simulations. The LES results agree well with the experimental results, but all our three simulations (3DSKE, 2DNLKE, and 3DNLKE) show different trends. The over prediction of stagnation-point value by the SKE model is a well-known fact and is attributed to the steep velocity gradient on the upstream side. The Figure 4.4 Streamline Patterns at Four Different Time 2DNLKE and 3DNLKE models predict Intervals for one Vortex Shedding Cycle the stagnation point correctly, but the maximum negative pressure and base pressure coefficient differ from the experimental results predicts the trend. Conclusions In the present work, 3-D unsteady computation of flow past a circular cylinder at subcritical Reynolds number has been performed using a nonlinear κ-ε model to evaluate its applicability. The same test case was simulated with 2DNLKE and its 3D counterpart 3DSKE to understand the effectiveness of the present model. The bulk parameters and the wake velocity recovery match well with experimental data and LES results. Since the grid requirement is not as severe as in LES and the number of cycles required to do averaging is also less, computational cost associated with the present model is very much less. For high-Re flows and flows encountered in practical engineering applications, there is a restriction on the mesh size and the LES technique is prohibitively expensive. Encouraging performance of the present NLKE model suggests that it could be used as an alternative
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tool in these situations. Further improvement in the prediction may be possible by making the model fully cubic form.
2nd Order Closures (Reynolds Stress Models - RSM) Due to the increased complexity of this class of turbulence models, 2nd order closure models do not share the same wide use as the more popular two-equation or algebraic models. The most noteworthy efforts in the development of this class of models was performed by [Donaldson and Rosenbaum (1968), Daly and Harlow (1970), and Launder , Reece, and Rodi (1975)]. The latter has become the baseline 2nd order closure model, with more recent contributions made by [Lumley (1978)], [Speziale (1985, 1987), and many other thereafter, who have added mathematical rigor to the model formulation. The first three belong to so called 1st order closures. LRR R-ε and LaunderGibson R-ε models, calculates the anisotropic Reynolds stresses present in typical flows of complex strain fields or significant body forces. RSM is the most complex of the turbulence models, and was designed to address the problems of the κ-ε model (which cannot predict flows in long non-circular duct because of the isotropic modelling of the Reynold’s stresses). The RSM can therefore accurately account for the Reynold’s stress field directional effects. Because of the many Reynold’s stresses to model, there are seven extra partial differential equations to solve, making computing costs very high. The Reynolds Stress models, sometimes called stress-equation models, are enforcing to those models which do not assume the Boussiqes assumption. Thus it would seems that these are perhaps the ultimate models. Nevertheless these models still utilize approximation and assumption in the modeling terms.
LES Model The principal idea behind LES is to reduce the computational cost by ignoring the smallest length scales, which are the most computationally expensive to resolve, via low-pass filtering of the Navier– Stokes equations. Such a low-pass filtering, which can be viewed as a timeand spatial-averaging, effectively removes small-scale information from the numerical solution. This information is not irrelevant, however, and its effect on the flow field must be modeled. A task which is an active area of research for problems in which small-scales can play an important role, such as near-wall flows , reacting flows, and multiphase flows49. Large-scale motion is calculated in LES, while the small-scale motion needs to Figure 4.5 Difference Between the Filtered Velocity be modeled because of the effects of largeand the Instantaneous Velocity scale motion. The most important aspect in application of LES is the use of suitable Sub-Grid Scale model. The implication to [Kolmogorov's (1941)] theory of self-similarity is that the large eddy of simulation are dependent on geometry, while the smaller scales more universal. This feature allows one to explicitly solve for the large eddies in a calculation and implicitly account for the small eddies by using a Sub-Grid-Scale (SGS) model. LES always solves three-dimensional, time dependent flow, calculating a mean of time-dependent flow fields. Therefore, it is best suited for transient simulations. The main difference between
49
Wikipedia.
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conventional turbulence modeling and LES is the averaging procedure. The LES technique does not involve the use of ensemble average; rather it consists in applying a spatial filter to N-S equations. Filter Definition Mathematically, one may think of separating the velocity field into a resolved and sub-grid part. The resolved part of the field represent the "large" eddies, while the sub-grid part of the velocity represent the "small scales" whose effect on the resolved field is included through the sub-grid-scale model50. This is called explicit filtering and Figure 4.5 illustrates the difference between the filtered velocity ūi and the instantaneous velocity ux. formally, one may think of filtering as the convolution of a function with a filtering kernel G:
ui (x⃗) = ∫ G(x⃗ − ξ)u(ξ)d(ξ)
ui = u̅i + u′i
Eq. 4.5 Where ūi the resolvable scale, and u’i is the sub-grid scale part. However, most practical (and commercial) implementations of LES use the grid itself as the filter (the box filter) and perform no explicit filtering. This is best shown in series of Error! Reference source not found. where L defines the largest eddies and Δ is the box filter based on mesh size. It is important to note that the large eddy simulation filtering operation does not satisfy the properties of a Reynolds operator. The expression most often used for sub-grid scale part as a geometric mean is
g ( x y z )1/3
Eq. 4.6
or its generalization, the cubic root of the cell volume. In case of anisotropic grids, definition tends to provide a fairly low value51. For this reason, the quadratic mean is used as following
2x 2y 2z g 3
1/2
Eq. 4.7
which is advocated in some publications. Other authors favor the maximum
g Max ( x , y , z )
Eq. 4.8
Filtered Incompressible N-S Equations Using Einstein notation, the Navier–Stokes equations for an incompressible fluid in Cartesian coordinates are:
u i 0 x i
,
u i u i u j 1 p 2ui ν x j x j ρ x i x jx j
Eq. 4.9
From Wikipedia. Jochen Frӧhlich , Dominic von Terzi,” Hybrid LES/RANS methods for the simulation of turbulent flows”, Progress in Aerospace Sciences 44 (2008) 349– 377. 50 51
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Filtering the Momentum equation results in
u i u i u j 1 p 2u i ν t x j ρ x i x jx j
or Eq. 4.10
u i u i u j 1 p 2 ui ν t x j ρ x i x jx j
This equation models the change of time of the filtered variable u͞i . Since the unfiltered variable ui are not known, it is impossible to directly calculate:
∂u ̅̅̅̅̅ i uj ⏟∂xj Not Known
≠
∂u̅i u̅j ⏟∂xj Known
Eq. 4.11 However the quantity on the right is known. Substituting:
∂u ̅̅̅̅̅ ∂u̅i u̅j ∂u̅i ∂u̅i u̅j 1 ∂p̅i ∂2 u̅i i uj + =− +ν − ( − ) ∂t ∂xj ρ ∂xi ∂xj ∂xj ⏟ ∂xj ∂xj where τij = u ̅̅̅̅̅ ̅ i u̅j i uj − u ⏟
∂τij ∂xj
SGS Modeling
Eq. 4.12 The result is a set of LES equations, resulting in one SGS turbulence model which can be formulated as the following52:
1 1 ∂u̅i ∂u̅j τij = −2μSGS S̅ij + τij δij where S̅ij = ( − ) 3 2 ∂xj ∂xi Eq. 4.13 Where SGS is the artificial or the sub-grid scale viscosity which acts as the constant of proportionality and Śij is the average strain rate. The size of the SGS eddies are determined by the filter choice as well as the filter cut-off width which is used during the averaging operation. The SGS viscosity can be obtained by the following semi-empirical formulation:
μSGS = ρ(CSGS ∆g )2 |√2S̅ij S̅ij | Eq. 4.14 where Δg as described before is the filter cut-off width, and the CSGS empirical constant which is usually specified in a range, 0.19 < CSGS < 0.24.53 52 T. Ganesan
and M. Awang, ”Large Eddy Simulation (LES) for Steady-State Turbulent Flow Prediction”, Springer International Publishing Switzerland 2015. 53 See Previuos.
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Numerical Methods for LES Large eddy simulation involves the solution to the discrete filtered governing equations using CFD. LES resolves scales from the domain size down to the filter size Δ and as such a substantial portion of high wave number turbulent fluctuations must be resolved. This requires either high-order numerical schemes, or fine grid resolution if low-order numerical schemes are used. [Pope]54 addresses the question of how fine a grid resolution is needed to resolve a filtered velocity field u(x). [Ghosal]55 found that for low-order discretization schemes, such as those used in finite volume methods, the truncation error can be the same order as the sub-filter scale contributions, unless the filter widths considerably larger than the grid spacing. While even-order schemes have truncation error, they are non-dissipative,56 and because sub-filter scale models are dissipative, even-order schemes will not affect the sub-filter scale model contributions as strongly as dissipative schemes. Implicit vs Explicit Filtering The filtering operation in large eddy simulation can be implicit or explicit. Implicit filtering recognizes that the sub-filter scale model will dissipate in the same manner as many numerical schemes. In this way, the grid, or the numerical discretization scheme, can be assumed to be the LES low-pass filter. While this takes full advantage of the grid resolution, and eliminates the computational cost of calculating a sub-filter scale model term, it is difficult to determine the shape of the LES filter that is associated with some numerical issues. Additionally, truncation error can also become an issue57. In explicit filtering, an LES filter is applied to the discretized Navier–Stokes equations, providing a well-defined filter shape and reducing the truncation error. However, explicit filtering requires a finer grid than implicit filtering, and the computational cost increases with [Sagaut]58 covers LES numeric in greater detail. [Sarkar and Voke]59 carried out an LES study of interactions of passing wakes and in flexional boundary layer over a low-pressure turbine blade. Figure 4.6 Vorticity Prompted by the Wake Figure 4.7 shows flow structures due to the complex Passing Cycle interactions of passing wakes and the separated shear layer where the iso-surface of vorticity at an instant of time through the wake passing cycle. Quantitative Aspects of Comparison of RANS vs. LES Models There are numerus studies done on comparison of advanced RANS models against large eddy
Pope, S. B. (2000). Turbulent Flows. Cambridge University Press. Ghosal, S. (April 1996). "An analysis of numerical errors in large-eddy simulations of turbulence". Journal of Computational Physics 125 (1): 187–206. 56 Randall J. Leveque (1992), “Numerical Methods for Conservation Laws (2nd Ed.)”, Birkhäuser Basel. ISBN 9783-7643-2723-1. 57 Grinstein, Fernando, Margolin, Len, Rider, William, “Implicit large eddy simulation”, Cambridge University Press. ISBN 978-0-521-86982-9, 2007. 58 Sagaut, Pierre, “ Large Eddy Simulation for Incompressible Flows”, (3rd Ed.), Springer, 2006. 59 “Large-eddy simulation of unsteady surface pressure over a LP turbine due to interactions of passing wakes and in flexional boundary layer” , Journal of Turbomachine, pp. 221–23, 2006. 54 55
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simulation (LES), chief among them is [Keshmiri et al.]60 etc. The biggest difference between LES and RANS is that, contrary to LES, RANS assumes that ūi = 0 (see the Reynolds-averaged Navier–Stokes equations). In LES the filter is spatially based and acts to reduce the amplitude of the scales of motion, whereas in RANS the time filter removes ALL scales of motion with timescales less than the filter width. What distinguishes LES from RANS is the definition of the small scales. While LES assumes the small scales to be smaller than the mesh size Δ, RANS assumes them to be smaller than the largest eddy scale L. So the quality of LES model is directly dependent in mesh size Δx. The mathematical similarity of LES and RANS equations, as evidence in the equations (4.15), are being solved essentially the same. However, the physics are different.
RANS LES
T u ( u u ) ν ν T u u p t T u (u u ) ν ν SGS u u p t
Eq. 4.15 It is obvious, the only change is in the dynamic viscosity determination, νT and νSGS. Or, the main difference being that in RANS the unclosed term is a function of the turbulent kinetic energy and the turbulent dissipation rate whereas in LES the closure term is dependent on the length scale of the numerical grid. So in RANS RANS the results are independent of the grid resolution! and usually the DES needs more refine mesh that RANS. Another point of view is that RANS can only give a time averaged mean value for velocity field since it is based on DES time averaging. In fact velocity field in this method is averaged over a time period of "Δt" which is considerably higher than time constant of velocity Figure 4.7 Time Averaged (RANS) vs Instantaneous (DES) fluctuations. An example would be the Simulation Over a Backup Step flow in backward step using both RANS and DES models, (see Figure 4.8) where the difference is obvious. While DNS resolves all scales of motion, all the way down to the Kolmogorov scale, LES is next up and resolves most of the scales, with the smallest eddies being modeled. RANS is on the other end of the spectrum from DNS, where only the large-scale eddies are resolved and the remaining scales are modeled. Motivations for Coupling Methods Between RANS and LES As indicated previously, (RANS) models provide results for mean quantities with engineering accuracy at moderate cost for a wide range of flows. In other situations, dominated by large-scale anisotropic vortical structures like wakes of bluff bodies, the average quantities are often less satisfactory when a RANS model is employed. Then large eddy simulation (LES) performs generally better and bears less modeling uncertainties. Furthermore, LES by construction provides unsteady data that are indispensable in many cases: determination of unsteady forces, fluid–structure coupling, identification of aerodynamic sources of sound, and phase-resolved multiphase flow, to Amir Keshmiri, Osman Karim, Sofiane Benhamadouche, “Comparison of Advanced Rans Models Against Large Eddy Simulation And Experimental Data In Investigation Of Ribbed Passages With Heat Transfer”, The 15th International Conference on Fluid Flow Technologies, Budapest, Hungary, September 4-7, 2012. 60
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name but a few issues [Frohlich & Terzi]61. Unfortunately, LES is by a factor of 10 to 100 more costly than RANS computations; LES requires a finer grid, cannot benefit from symmetries of the flow in space, and provides mean values only by averaging the unsteady flow field computed with small time step over a long sampling time. Hence, it seems natural to attempt a combination of both turbulence modeling approaches and to perform LES only where it is needed while using RANS in regions where it is reliable and efficient. Another and somewhat different motivation for LES/RANS coupling stems from wall-bounded flows. Close to walls, the LES philosophy of resolving the locally most energetic vortical structures requires to substantially reduce the step size of the grid since the dominating structures become very small in this region. Furthermore, when increasing the Reynolds number, the scaling of the computational effort is similar to that of a DNS in its dependence on Re just with a smaller constant. That makes the approach unfeasible for wall-bounded flows at high Re, such as the flow over a wing . As a remedy, some sort of wall model can be introduced to bridge the near-wall part of the boundary layer and to make the scaling of the required number of grid points independent of Re. A RANS model depends on physical quantities describing the entirety of the turbulent fluctuations. For the sequel it is necessary to define the specifics of LES models and RANS models. Using an unsteady definition of a Reynolds average as discussed above, the transport equations for the Reynolds-averaged velocity read:
ui t
ui u j x j
p ν u i x j x j x j
τ ijRANS x j
DES u i u i u j p νu i τ ij t x j x j x j x j x j
Eq. 4.16
For example, the κ–ε model determines
ui τ ijRANS f , κ, ε, C x i
u i , τ ijDES f , Δ g , C x i
Eq. 4.17
where C is a model constant, κ the turbulent kinetic energy, and ε the turbulent dissipation rate. The latter two are determined from other relations. For LES based on the (Smagorinsky) model uses a relation like where Δg is a length scale related to the numerical grid, Since there exist many variants of LES and RANS models we define the following: a model qualifies as an LES model if it explicitly involves in one or the other way the step size of the computational grid. RANS models, in contrast, only depend on physical quantities, including geometric features like the wall distance. Principal Approaches to Coupling LES with RANS The similarity of the equations and the considerations suggest the concept of unified modeling. This approach is based on using the same transport equation for some resolved velocity ūi, yet to be specified in its meaning:
Jochen Frӧhlich , Dominic von Terzi,” Hybrid LES/RANS methods for the simulation of turbulent flows”, Progress in Aerospace Sciences 44 (2008) 349– 377. 61
50
Model u i u i u j p νu i τ ij t x j x j x j x j x j
Eq. 4.18
A transition from LES to RANS can be achieved in several ways. One possibility is blending, i.e. by a weighted sum of a RANS model and an LES the models according to
ijModel f RANS ijRANS f LES ijLES
Eq. 4.19
In this fashion, f RANS and f LES are local blending coefficients determined by the local value of a given criterion. Another strategy is to use a pure LES model in one part of the domain and a pure RANS model in the remainder, so that a boundary between a RANS zone and an LES zone can be specified at each instant in time. The transport equation for the velocity, however, is the same in both zones with no particular adjustment other than switching the model term at the interface. This way the computed resolved velocity is continuous. We term this strategy Interfacing LES and RANS. Furthermore, if the interface is constant in time, it is called a hard interface. If it changes in time depending on the computed solution, it is termed a soft interface. Segregated modeling is the counterpart to unified modeling as LES is employed in one part of the computational domain, while RANS is used in the remainder. With segregated modeling, however, the resolved quantities are no more continuous at the interfaces. Instead, almost stand-alone LES and RANS computations are performed in their respective subdomains which are then coupled via appropriate boundary conditions. Except for laminar flows, the solution is discontinuous at these interfaces. This avoids any gradual transition in some gray area characteristic of unified turbulence models. Segregated modeling allows for embedded LES by designing a configuration where in an otherwise RANS simulation a specific region is selected to be treated with LES with full two-way coupling between the zones62. Results and Discussion for Segregated Modeling As an example of segregated modeling, is integrated RANS-LES of the NASA stage 35 compressor and a diffuser. While the compressor stage is computed with the RANS flow solver, the subsequent diffuser with the LES flow solver (see Figure 4.9). Here, we look at the vorticity magnitude distribution at the 50% clip plane of the stator. Again, we can identify the wakes of the stator passing the interface. The different description of turbulence in the two mathematical approaches is apparent. While on the RANS side the turbulence is modeled in a turbulence model and cannot be seen in the Figure 4.8 Integrated RANS-LES Computations in Gas vorticity distribution, on the LES side Turbines: Compressor-Diffuser, Jochen Frӧhlich , Dominic von Terzi,” Hybrid LES/RANS methods for the simulation of turbulent flows”, Progress in Aerospace Sciences 44 (2008) 349– 377. 62
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the fine scale turbulence is regenerated and can be identified as small-scale structures in the LES solution63. Another example in the feasibility of a hybrid RANS–LES approach is the numerical simulation of aircraft wing-tip vortices which has been studied by [Kolomenskiy et al]64. Mesh sensitivity tests of our RANS solver and comparisons between two different turbulence models indicate that the RANS approach adequately describes the flow upstream from the trailing edge, but overestimates the rate of decay of the wing-tip vortex. A hybrid RANS–LES method is presented that results in a better agreement with the wind tunnel experiment for numerical simulation of the wake of an airliner. Coupling in RANS/LES Interface A fully coupled solution requires that all flow variables must be exchanged at the interface. When some engine components are computed with LES and others with RANS approximations will have to be made to couple instantaneous and averaged variables. To simplify the problem, we consider only the one-way coupling of the velocity and turbulence variables. One-way coupling means that information is passed only downstream; the variables at the inlet of the downstream domain are computed from the variables at the outlet of the upstream domain. For the RANS/LES interface, turbulent fluctuations need to be added to the velocity from an upstream RANS computation. For the LES/RANS interface, such as the combustor/turbine interface, a simple time average of the velocity provides a mean velocity at the inlet of the RANS domain. This velocity distribution is again highly nonuniform, which allows to describe turbulence at the inlet with the local turbulence generation from the mean velocity. The flow in the compressor is computed with unsteady RANS using the k-ω model and the flow in the combustor is computed with LES. The last blade row of the compressor is a row of stators and does not rotate. The blade row upstream is a row of rotors that rotate counter-clockwise. Figure 4.9 Compressor and combustor: RANS and LES axial The mean flow is highly complex, velocity, mid-passage (Courtesy of Medic et al.) with the wakes originating from the last stage of the compressor (see Figure 4.10). These wakes are unsteady due to the rotation of the rotors in the compressor. Larger values of turbulent kinetic energy are in the regions with strong velocity gradients. The large values of k near the hub might be spurious; they are highly dependent on the quality of the grid. This illustrates the fact k from the k-ω model usually fails to accurately represent the true turbulent kinetic energy in complex flows. The flow field in the downstream LES domain is highly dependent on the conditions at the inlet. To generate an inflow profile for the LES in the combustor, the mean velocity at the combustor inlet is set equal to the RANS velocity at the compressor exit and appropriate fluctuations need to be added [Medic, et al]65. J. U. Schlüter, X. Wu, S. Kim, J. J. Alonso, and H. Pitsch, AIAA-2004-369, 42nd Aerospace Sciences Meeting and Exhibit Conference, January 2004. 64 Dmitry Kolomenskiy, Roberto Paoli, and Jean-Franc¸ois Boussuge, “Hybrid Rans–Les Simulation of Wingtip Vortex Dynamics”, FEDSM2014-21349. 65 G. Medic, D. You AND G. Kalitzin, “An approach for coupling RANS and LES in integrated computations of jet engines”, Center for Turbulence Research, Annual Research Briefs, 2006. 63
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The aim of segregated modeling is to compute all models in their regime of validity: steady RANS for flows with stationary statistics and unsteady LES with high resolution where it is needed. Therefore one can choose the best suited method for each subdomain without considering their compatibility and without fear of inconsistencies in their use. Furthermore, any gray zone where the model is left alone with generating fluctuations in some transition process is avoided. The price to pay is the need for comparatively complex coupling conditions. For block-structured solvers, however, the routines for data exchange required anyway facilitate a straightforward implementation. Inappropriate coupling conditions lead to contamination of the results in the LES or RANS subdomains. Depending on the type of the interface, the requirements on the coupling conditions (inlet and outlet) differs [Frohlich & Terzi]66
DES Model
An alternative between LES and RANS models would the Detached Eddy Simulations (DES) which attempts to combine the best aspects of RANS and LES methodologies in a single solution strategy. This is more pronounce in near-wall regions in a RANS-like manner, and treat the rest of the flow in an LES-like manner. The model was originally formulated by replacing the distance function d in the Spalart-Allmaras (SA) model with 13 M cells a modified distance function D as:
D Min (d, CDES , ) Eq. 4.20 Where CDES is a constant and Δ is the largest dimension of the grid cell in question. This modified distance function causes the model to behave as a RANS model in regions close to walls, and in a Smagorinsky-like manner (LES) away from the walls. This is 16 M cells usually justified with arguments that the scale-dependence of the model is made local rather than global, and that dimensional analysis backs up this claim. In summary, an example of a hybrid technique, detached eddy simulation (DES) is a modification of a RANS model in which the model switches to a sub-grid scale formulation in regions fine enough for LES calculations. Therefore, the grid resolution is not as demanding as Figure 4.10 DES and Effect of Grid Density on the Wake Flow pure LES, there by considerably cutting down the cost of the computation. A study in CD Adapco® shows that for wake flow drag calculation, DES is more accurate Jochen Frӧhlich , Dominic von Terzi,” Hybrid LES/RANS methods for the simulation of turbulent flows”, Progress in Aerospace Sciences 44 (2008) 349– 377. 66
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than RANS and highly dependent in grid resolution (see Figure 4.11, top 13 M cells vs bottom-16 M cells). Another example would be the presentation of flow in rectangular ogive fore-body (Aircraft fore body)67. In that environment we set close to the wall D is by the wall parallel spacing’s, i.e., D = d; d =
