Dean of the Graduate College ..... Top-Wall Deflection angle from Driver and ..... housings for automotive air filters and headers for compact cross flow heat ...
NUMERICAL ANALYSIS OF BACKWARD-FACING STEP FLOW PRECEEDING A POROUS MEDIUM USING FLUENT
By CHANDRAMOULEE KRISHNAMOORTHY Bachelor of Engineering University of Mumbai (Bombay) Mumbai, India 2004
Submitted to the Faculty of the Graduate College of the Oklahoma State University in partial fulfillment of the requirements for the Degree of MASTER OF SCIENCE December, 2007
NUMERICAL ANALYSIS OF BACKWARD-FACING STEP FLOW PRECEEDING A POROUS MEDIUM USING FLUENT
Thesis Approved: Frank W. Chambers David G. Lilley Afshin J. Ghajar A. Gordon Emslie Dean of the Graduate College
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TABLE OF CONTENTS
1.INTRODUCTION ........................................................................................................... 1 1.1 Background ............................................................................................................... 1 1.2 Backward Facing Step .............................................................................................. 2 1.3 Objectives ................................................................................................................. 4 2.REVIEW OF LITERATURE .......................................................................................... 5 2.1 Introduction............................................................................................................... 5 2.2 Experimental Studies on Backward Facing Steps .................................................... 6 2.3 Numerical Studies on Backward Facing Steps ....................................................... 14 2.3.1 Laminar Flow Analyses: 2D and 3D ........................................................... 14 2.3.2 Direct Numerical Simulation (DNS) ........................................................... 17 2.3.3 Large Eddy Simulation (LES) ..................................................................... 19 2.3.4 Turbulent Flow Analysis: Modeling of RANS............................................ 19 2.4 Experimental Studies on Porous Media.................................................................. 21 2.5 Turbulent Flow modeling in Porous media ............................................................ 24 2.5.1 Space and Time Approach ........................................................................... 25 2.5.2 Time and Space Approach ........................................................................... 28 2.6 Backward Facing Step and Porous Media .............................................................. 32 2.7 Previous Work at OSU............................................................................................ 36 2.8 Conclusions of the Review ..................................................................................... 38 3.NUMERICAL APPROACH ......................................................................................... 39 3.1 Introduction............................................................................................................. 39 3.2 Governing Equations .............................................................................................. 41 3.2.1 Clear Fluid Region....................................................................................... 41 3.2.2 Porous Region.............................................................................................. 42 3.2.3 Boundary Condition at the Interface of Clear Fluid and Porous Media ...... 44 3.3 Turbulence Models in FLUENT............................................................................. 44 3.3.1 Spalart – Allmaras (SA)............................................................................... 45 3.3.2 Standard k-ε (SKE) ...................................................................................... 45 3.3.3 Renormalization Group k-ε (RNG).............................................................. 45 3.3.4 Realizable k-ε (RKE) ................................................................................... 46 3.3.5 Standard k-ω (SKW).................................................................................... 46 3.3.6 Shear Stress Transport k-ω (SST)................................................................ 47
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Chapter
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3.3.7 Reynolds Stress Model (RSM) .................................................................... 47 3.4 Grid Generation in GAMBIT.................................................................................. 48 3.4.1 Turbulent Boundary Layer........................................................................... 49 3.4.2 Wall Function Approach.............................................................................. 51 3.4.3 Damping Function Approach....................................................................... 51 3.4.4 Two Layer Model Approach........................................................................ 52 3.4.5 Determination of Distance of First Grid Point from the Wall ..................... 52 3.4.6 Samples of Final Grids................................................................................. 53 3.5 Simulation in FLUENT .......................................................................................... 55 4.RESULTS AND DISCUSSION .................................................................................... 60 4.1 Grid Independence Studies ..................................................................................... 60 4.1.1 Grid Independence: Armaly et al. (1983) .................................................... 61 4.1.2 Grid Independence: Yao et al. (2000).......................................................... 63 4.2 Numerical Results from FLUENT.......................................................................... 65 4.2.1 Re = 2000 and Re = 3750 ............................................................................ 65 4.2.2 Re = 6550 and Re =10000 ........................................................................... 72 4.2.3 Separation Lines........................................................................................... 80 4.2.4 Effect of Variation of Permeability, Inertial Constant and Thickness on Separation Lines........................................................................................... 85 5.CONCLUSIONS AND RECOMMENDATIONS ........................................................ 88 5.1 Conclusions............................................................................................................. 88 5.2 Recommendations................................................................................................... 89 REFERENCES ................................................................................................................. 91 APPENDIX A SNAPSHOTS FROM FLUENT 6.1 ...................................................... 99 APPENDIX B RESULT TABLES............................................................................... 104
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LIST OF TABLES
I. Geometries of Experiments Validated in the Present Study................................... 3 II. Comparison of Experimental and Numerical approaches of Silveira et al. (1991).............................................................................................. 19 III. Flow Regimes in Porous Media Summarized from Experiment by Dybbs and Edwards (1984)................................................................................... 23 IV. Physical Properties of Air at 20 oC ....................................................................... 56 V. Boundary Conditions for Backward Facing Step Geometry ................................ 56 VI. Input Values for Velocity Inlet Boundary Condition ........................................... 57 VII. Input Values for Porous Jump Boundary Conditions ........................................... 58 VIII. Various Grid Sizes Used for Realizable k-ε Model at Re = 7000 ........................ 62 IX. Various Turbulence Models Used for Geometry of Armaly et al. (1983) at Re = 7000 .......................................................................................................... 64 X. Various Grid Sizes Used for Geometry of Yao (2000); Realizable k-ε Model at Re = 6550 .......................................................................................................... 65 XI. Re-attachment Lengths at Different Reynolds Numbers Using Realizable k-ε Model .............................................................................................................. 65
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LIST OF FIGURES
1-1: Schematic of Two-dimensional Backward Facing Step.…………….…………….... 3 2-1: Road-map for Literature Review................................................................................. 5 2-2: Comparison of Experimental and Numerical Results of Armaly et al. (1983), from Kanna and Das (2006); x1: Re-attachment Length (XR) and step: Step height (h)............................................................................................................ 7 2-3: Re-Attachment location vs. Top-Wall Deflection angle from Driver and Seegmiller (1985); x: Re-attachment Length (XR) and H: Step height (h)................ 8 2-4: Schematic of Wind-Tunnel Section from Jovic and Driver (1994) ............................ 9 2-5: Re-Attachment Length vs. Reynolds Number from Lee and Mateescu (1998) ■: Lee and Mateescu (1998); ○: Goldstein et al. (1970); ●: Armaly et al. (1983); xr: Re-attachment length (XR); H: Step height (h) ...................................... 10 2-6: Mean Velocity Flow Field Obtained from PIV, from Pilloni et al. (2000); U0: Maximum Velocity at the step (Umax)...................................................................... 11 2-7: Secondary Recirculation Region Obtained from PIV, from Hall et al. (2003) ......... 12 2-8: (a) Re-attachment Length vs. Reynolds Number (b) Re-attachment vs. Channel Span from Beaudoin et al. (2004); LR: Re-attachment Length (XR) ......... 13 2-9: Re-Attachment Length vs. Reynolds Number for Laminar Flow from Kim and Moin (1985); xr: Re-attachment length (XR)..................................................... 16 2-10: Skin Friction Co-efficient vs. x/h from Bredberg et al. (2002) .............................. 21 2-11: Cross section of Porous Medium Packed Bed of Spheres, from Dybbs and Edwards (1984)........................................................................................................ 23 2-12: Schematic of Pseudo and Void Vortices from Masuoka and Takatsu (1996)......... 29 2-13: Sensitivity of Flow Field (stream-traces) to Changes in Darcy Number for b/h = 0:3 and F = 0:55 from Chan and Lien (2005)................................................. 33
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Figure
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2-14: Sensitivity of Flow Field (stream-traces) to Changes in the Forchheimer’s Constant for b/h = 0:3 and Da = 0:01 from Chan and Lien (2005) ......................... 33 2-15: Sensitivity of Flow Field to changes in the Thickness of Porous Insert for Da = 0:01 and F = 0:1 from Chan and Lien (2005) ................................................. 34 2-16: Comparison of Streamlines Between the Linear and Nonlinear Models for Backward-facing-step Flow with Porous Insert, α = 10–6 m2, φ = 0.65 from Assato et al. (2005) ......................................................................................... 35 2-17: Comparison of Streamlines Between the Linear and Nonlinear Models for Backward-facing-step Flow with Porous Insert, α = 10–6 m2, φ = 0.85 from Assato et al. (2005) ......................................................................................... 35 2-18: Comparison of Streamlines Between the Linear and Nonlinear Models for Backward-facing-step Flow with Porous Insert, α = 10–7 m2, φ = 0.85 from Assato et al. (2005) ......................................................................................... 36 3-1: The CFD Simulation Pipeline for Fluent Preprocessing-2006 (Fluent Inc.)............. 41 3-2: Boundary Conditions of the Edges in GAMBIT....................................................... 41 3-3: Turbulent Boundary Layer Profile in the Near-wall Region..................................... 50 3-4: Reτ as a Function of Reynolds number from Pope (2000) …………………………53 3-5: Sample Structured Grid of Armaly et al. (1983) ....................................................... 54 3-6: Sample Structured Grid of Yao (2000) – No Filter case ........................................... 54 3-7: Sample Clustered Grid of Yao (2000) – Filter at 4.25 Step Heights......................... 54 4-1: Separation Lines at Re = 2000: FLUENT ................................................................. 67 4-2: Comparison of Experiment and FLUENT: No Filter Case, Re = 2000 .................... 68 4-3: Comparison of Experiment and FLUENT: Filter at 4.25 h, Re = 2000 .................... 68 4-4: Comparison of Experiment and FLUENT: Filter at 6.75 h, Re = 2000 .................... 69 4-5: Separation Lines at Re = 3750: FLUENT ................................................................. 70 4-6: Comparison of Experiment and FLUENT: No Filter Case, Re = 3750 .................... 70
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Figure
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4-7: Comparison of Experiment and FLUENT: Filter at 4.25 h, Re = 3750 .................... 71 4-8: Comparison of Experiment and FLUENT: Filter at 6.75 h, Re = 3750 .................... 71 4-9: Separation Lines at Re = 6550: FLUENT ................................................................. 72 4-10: Comparison of Experiment and FLUENT: No Filter Case, Re = 6550 .................. 74 4-11: Comparison of Experiment and FLUENT: Filter at 4.25 h, Re = 6550 .................. 74 4-12: Comparison of Experiment and FLUENT: Filter at 6.75 h, Re = 6550 .................. 75 4-13: FLUENT Velocity Profiles at 3.75 h; With and Without Filter at 4.25 h, Re = 6550................................................................................................................. 75 4-14: FLUENT Velocity Profiles at 5h; With and Without Filter at 6.75 h, Re = 6550................................................................................................................. 76 4-15: Separation Lines at Re = 10000: FLUENT ............................................................. 77 4-16: Comparison of Experiment and FLUENT: No Filter case, Re = 10000 ................. 77 4-17: Comparison of Experiment and FLUENT: Filter at 4.25 h, Re = 10000 ................ 78 4-18: Comparison of Experiment and FLUENT: Filter at 6.75 h, Re = 10000 ................ 79 4-19: FLUENT Velocity Profiles at 3.75 h: With and Without Filter at 4.25 h, Re = 10000............................................................................................................... 79 4-20: FLUENT Velocity Profiles at 6.25 h: With and Without Filter at 6.75 h, Re = 10000............................................................................................................... 80 4-21: Separation Lines for No Filter Case: Yao (2000).................................................... 81 4-22: Separation Lines for No Filter Case: FLUENT....................................................... 81 4-23: Separation Lines for Filter at 4.25 h: Yao (2000) ................................................... 83 4-24: Separation Lines for Filter at 4.25 h: FLUENT ...................................................... 83 4-25: Separation Lines for Filter at 6.75 h: Yao (2000) ................................................... 84 4-26: Separation Lines for Filter at 6.75 h: FLUENT ...................................................... 84
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Figure
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4-27: Effect of Variation of Permeability (α) on Separation Lines by FLUENT; Inertial Constant (C2) = 4.533*103 1/m, Thickness (b) = 15 mm for Filter Placed at 4.25 h: Re=10000………………………………………………………..86 4-28: Effect of Variation of Inertial Constant (C2) on Separation Lines by FLUENT; Permeability (α) = 1.17*10-9 m2, Thickness (b) = 15 mm for Filter Placed at 4.25 h: Re=10000 ..................................................................................... 86 4-29: Effect of Variation of Thickness (b) on Separation Lines by FLUENT; Permeability (α) = 1.17*10-9 m2, Inertial Constant (C2) = 4.533*103 1/m for Filter Placed at 4.25h: Re=10000.……………………….………….……….....87
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NOMENCLATURE
b
Thickness of porous inserts
Cf
Skin friction coefficient
Cε
Coefficient determined from experiment
C2
Pressure jump coefficient
dP
Characteristic length of pore
Dh
Hydraulic diameter
Da
Darcy number
F
Forchheimer’s constant
F
Body force vector
g
Gravity vector
h
Step height
I
Identity matrix
k
Kinetic energy
p
Pressure
Re
Reynolds number
Reh
Reynolds number based on step height
ReP
Pore Reynolds number
Reτ
Reynolds number based on uτ
s
Channel height
x
t
Time
ui
Velocity in primary direction
ui
Mean velocity in primary direction
ui′
Velocity fluctuation in primary direction
u+
Dimensionless wall velocity
uτ
Friction velocity
Ubulk
Bulk velocity of the flow
Uinlet
Boundary condition input at the inlet of channel
Umax
Maximum velocity at step
UP
Fluid velocity through pore
U
Mean velocity
v
Velocity
X
Length in X-direction
x1
Re-attachment length
xi
Primary direction
xj
Other direction
XR
Re-attachment length
Y
Length in Y-direction
yP
Actual distance from the wall
y+
Wall units
z
Width in Z-direction
α
Permeability of the filter
δij
Kronecker delta xi
∆p
Pressure drop
ε
Turbulence dissipation
κ
Log-law coefficient
ϕ
Porosity of the filter
ρ
Fluid density
τ
Stress tensor
µ
Dynamic viscosity
ν
Kinematic viscosity
νT
Turbulent viscosity
ω
Specific dissipation rate
∇
Del operator
ABBREVIATIONS
AR
Aspect Ratio
CFD
Computational Fluid Dynamics
DES
Detached Eddy Simulation
DNS
Direct Numerical Simulation
ER
Expansion Ratio
LDA
Laser Doppler Anemometer
LDV
Laser Doppler Velocimeter
LES
Large Eddy Simulation
LIF
Laser Induced Fluorescence
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N-S
Navier Stokes
PDE
Partial Differential Equation
PIV
Particle Image Velocimetry
PVC
Poly Vinyl Chloride
RANS
Reynolds Averaged Navier Stokes
RNG
Renormalization Group
RSM
Reynolds Stress Model
SA
Spalart Allmaras
SKE
Standard k-ε
SKW
Standard k-ω
SST
Shear Stress Transport k-ω
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CHAPTER 1
INTRODUCTION
1.1 Background One of the important engineering applications, where the fluid flows through unexpected bends and encounters sudden expansions is the ‘air filter housings’ of automobiles. The rationale behind this complex flow path in present day automobiles is that the design criteria are determined by space utilization rather than fluid mechanics. This arrangement results in the flow being non-uniform and the mean velocity of the flow is not normal to the surface of the filter. Moreover, the velocity fluctuations observed in the separation region combined with non-uniform flow are found to be detrimental to the performance of the filter. Previous research has shown that the velocity fluctuations and non-uniform flow through the filter are the important factors on which the filtration efficiency depends. The real flow field through air filter housing, when considered with all its geometrical parameters is extremely intricate and expensive to simulate numerically. Also, it is not feasible to measure all the minute details of the real flow field. Thus, the need to build a simplified model of this complex flow that will result in a better understanding of the interaction between the separation region and the porous medium is highlighted in this research. For some engineering applications, the above phenomena (separation and re-attachment) facilitates in enhancing the momentum, heat and/or mass transfer rates while for the others, it may lead to an unsteady flow resulting
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in noise, vibration and reduced efficiency. Other significant applications of flow modeling through porous media can be found in designs of fluidized bed combustors, catalytic reactors, crude-oil drilling, flows in the core of nuclear reactors and environmental flows over forests and vegetation. Hence, turbulent flow modeling in porous media is an essential exercise in understanding various complex engineering and environmental flows.
1.2 Backward Facing Step The backward-facing step flow (see Figure 1-1) is a fundamental flow that provides a simple geometry to serve as a prototype for studying complex phenomena like flow separation and re-attachment. It is similar to many industrial flows, including housings for automotive air filters and headers for compact cross flow heat exchangers. A comprehensive understanding of these phenomena is of prime importance for the design of engineering devices like diffusers, turbines, combustors, airfoils, etc. Figure 1.1 shows a two-dimensional schematic of a backward facing step with a porous insert. The channel height is denoted by‘s’ and step-height is denoted by ‘h’. The expansion ratio (ER) is then defined as shown in Equation 1.1.
ER =
s s+h
(1.1)
In the present study the step-flow experiments of Armaly et al. (1983) and Yao (2000) were modeled using commercial software FLUENT and the mesh was created
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using pre-processing software GAMBIT. Table I details the geometries of the two experimental studies examined in the present study.
Umax
Filter s
Inflow
Dividing Streamline
Y
Outflow
h X Re-attachment Length (XR)
Figure 1-1: Schematic of Two-dimensional Backward Facing Step
Table I: Geometries of Experiments Validated in the Present Study Geometry
Expansion Ratio (ER)
Aspect Step Height (h) Channel Height (s) Ratio (AR) mm mm
Armaly et al. (1983)
1 : 1.94
4.9
5.2
1: 36
Yao (2000)
1: 2
25
25
1: 20
In the present study, the Reynolds number (Re) is based on the hydraulic diameter (Dh = 2s) of the inlet channel and Bulk velocity (Ubulk) at the step, as shown in Equation 1.2. This nomenclature of the Reynolds number is consistent with those of Armaly et al. (1983) and Yao (2000).
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Re =
ρ U bulk Dh ρ (0.857 × U max ) Dh ρ (1.024 × U inlet ) Dh = = µ µ µ
(1.2)
Various turbulence models including: Spalart-Allmaras, different versions of k-ε and k-ω models and, the Reynolds Stress Models are available in FLUENT. The present study utilizes these options to investigate the performances of different turbulence models when applied to intricate flows involving separation and porous media.
1.3 Objectives The objectives of the present study were: •
No filter case: To compare the numerical simulations (using FLUENT software) with the experiments of Yao (2000) at Re = 2000, 3750, 6550 and 10000.
•
Flow with filter: To compare the numerical simulations (using FLUENT) with the experiments of Yao (2000) at Re = 2000, 3750, 6550 and 10000 with the filter placed at 4.25 and 6.75 step heights from the step.
•
To obtain the effect of varying filter parameters such as permeability, inertial constant and media thickness on the dynamics of the flow.
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CHAPTER 2
REVIEW OF LITERATURE
2.1 Introduction The organization of the chapter is shown as a road-map given in Figure 2-1 It can be seen that the review discusses prominent experimental and numerical studies on backward facing step. Moreover, experimental and numerical studies on turbulence flow in porous media also are discussed. Finally, conclusions of the review are presented.
Backward Facing Step
Experimental Studies Numerical Analyses
2-D and 3-D Laminar Analyses Direct Numerical Simulation (DNS) Large Eddy Simulation (LES)
Turbulence in porous media
Experimental Studies Numerical Studies
RANS Modeling
Space and Time Approach Time and Space Approach
Conclusions of the review
Figure 2-1: Road-map for Literature Review
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2.2 Experimental Studies on Backward Facing Steps Significant interest was revived in the early 1980’s in the flow over backward facing steps with the experiments of Durst and Tropea (1981), Sinha et al. (1981) and Armaly et al. (1983). Durst and Tropea (1981) found experimentally the effect of expansion ratio and Reynolds number on the re-attachment length. The authors found that the re-attachment length increases with both Expansion Ratio (ER) and Reynolds number. Their experimental results with ER = 20 were similar to those of Eaton and Johnston (1980) who used a channel of ER = 16.6. Sinha et al. (1981) experimentally analyzed both laminar and turbulent regimes over backward-facing step. The range of Reynolds numbers investigated in their experiment was from 100 to 12500. Their results showed that the re-attachment length linearly increased in the laminar regime (till Re = 800); then drops as Reynolds number increases and finally reaches a constant value of around six step heights (6h) for Re > 10000. Armaly et al. (1983) investigated the effect of Reynolds Number on the reattachment length. The range of Reynolds number over which the experiments were performed covered all the three flow regimes. Additional separation regions were found on the non-step side of the step which was previously never reported in any literature. Moreover, the flow over the step showed signs of two dimensional behavior only at very low and very high Reynolds number (Re < 400 and Re > 6600) and was mainly three dimensional for Reynolds numbers between the above ranges. Their investigations led to the conclusion that at (a) at low Reynolds number (Re < 1200): the re-attachment length increased with Reynolds number (b) for the transitional flow regime (1200 < Re < 6600):
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the re-attachment length decreased slightly and (c) for the fully turbulent flow it remains relatively constant. Moreover, the experiments of Durst and Tropea (1981) and Sinha, et al. (1981) were in good agreement with Armaly et al. (1983). Figure 2-2 shows the reattachment results for the laminar flow regime of Armaly et al. (1983).
Figure 2-2: Comparison of Experimental and Numerical Results of Armaly et al. (1983), from Kanna and Das (2006); x1: Re-attachment Length (XR) and step: Step Height (h)
Driver and Seegmiller (1985) analyzed the effect of pressure gradient on reattachment length of turbulent flow over a backward facing step. They varied the pressure gradient by changing the angle of the top-wall with the horizontal while velocity measurements were carried out with Laser Doppler Anemometer (LDA). The authors found an increase in reattachment length with top-wall angle. They also compared the numerical predictions with the experimental results and found that the numerical results under-predict the reattachment length. Their results can be observed from Figure 2-3.
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Figure 2-3: Re-Attachment Location vs. Top-Wall Deflection Angle from Driver and Seegmiller (1985); x: Re-attachment Length (XR) and H: Step Height (h) Adams and Eaton (1988) measured the velocity profiles and skin-friction coefficient over the backward facing step by a single component LDA. The experiment showed the importance of upstream initial flow conditions on the development of the free shear layer. The authors noted that a thick boundary layer caused a lower pressure rise to re-attachment and a lower pressure gradient at re-attachment, than the cases with thinner initial separating boundary layers. Isomoto and Honami (1989) investigated the effect of turbulent intensity on reattachment length. The authors found evidence of negative correlation between reattachment length and maximum turbulence intensity near the wall. Moreover, they comment that the re-attachment length is significantly affected by the turbulence in the re-circulation region directly below the step.
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Figure 2-4: Schematic of Wind-Tunnel Section from Jovic and Driver (1994)
The Laser-Oil flow interferometry technique was used by Jovic and Driver (1994) and Jovic and Driver (1995) to measure the shear stress at the walls of a backward facing step. They comment that this technique is more robust than other conventional methods due to its non-dependence on the law of the wall. The experiment (See Figure 2-4) revealed that skin-friction coefficient decreased everywhere in the flow field as the Reynolds number increased. This behavior is observed both in the recirculation region as well as in the recovery region. By measuring the skin-friction in the recirculation region, they found that the flow near the wall in the recirculation region behaves like a viscousdominant laminar like flow. Lee and Mateescu (1998) analyzed laminar and transitional flows by conducting experiments on steps with ER’s of 1.17 and 2.0. The authors also performed numerical investigations and found both the results in agreement with prior studies. Figure 2-5 shows the re-attachment length as a function of Reynolds number.
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Benedict and Gould (1998) found a re-attachment length of 6.38h for a Reynolds number of Reh = 23600. (h = step height). The expansion ratio of the channel was 1.25. The authors observed that the velocity profiles in the re-circulation region did not obey the log-law profile in the near wall region. Moreover, they exhibited the same behavior until a distance of 2.1XR (XR = Re-attachment length) after the re-attachment point.
Figure 2-5: Re-Attachment Length vs. Reynolds Number from Lee and Mateescu (1998) ■: Lee and Mateescu (1998); ○: Goldstein et al. (1970); ●: Armaly et al. (1983); xr: Re-attachment length (XR); H: Step Height (h)
Pilloni et al. (2000) employed PIV and LDA over a backward facing step in order to compare the two flow measurement techniques. They confirmed that using PIV and LDA together would enable the researchers to exploit the advantages of both systems and 10
complete spatial and temporal information of the flow field can be obtained (see Figure 2-6).
Figure 2-6: Mean Velocity Flow Field Obtained from PIV, from Pilloni et al. (2000); U0: Maximum Velocity at the Step (Umax)
Armaly et al. (2003) investigated the step flow in the Reynolds range of 98.5 < Re < 525. The authors observed a span-wise velocity distribution near the step wall. Moreover, they observed that the stream-wise velocity distribution reaches a maximum value (in the span-wise direction) near the wall not on the centre-line of the span. Hall et al. (2003) analyzed the secondary corner vortex in the re-circulation zone using 2-D cross correlation PIV measurements. Figure 2-7 shows a magnified PIV plot of streamlines near the step-wall. They found that similar to the primary vortex, the secondary vortex was highly three dimensional in nature. Hence the authors doubt the two dimensional assumption of flow structures even for high aspect ratio steps. The measurements were performed for Reh = 44000.
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Figure 2-7: Secondary Recirculation Region Obtained from PIV, from Hall et al. (2003) Piirto et al. (2003) conducted turbulence energy budgets for a backward facing step with an expansion ratio of 1.5. The Reynolds number (Reh) in this study was 15000 and flow measurements were performed using 3-Component Particle Image Velocimetry (PIV). However, the turbulence energy budgets obtained from PIV data were almost double that of DNS results of Le et al. (1997). The authors cannot explain this unusually high energy content. Beaudoin et al. (2004) found evidence of three-dimensional stationary structures that vary periodically in the span-wise direction. However, their numerical results are not consistent with their experimental results. The re-attachment length found by numerical simulation is 7h while experiments show it to be 4.5h. The span-wise periodic variation for Reynolds numbers (Re) varying from 0 to 300, is shown in Figure 2-8. It also shows
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the variation of experimental re-attachment length with Reynolds number and its comparison to the results of Armaly et al. (1983).
Figure 2-8: (a) Re-attachment Length vs. Reynolds number (b) Re-attachment vs. Channel Span from Beaudoin et al. (2004); LR: Re-attachment Length (XR)
Hudy et al. (2005) found that the re-attachment point increases and secondary vortex reduces in size as the Reynolds number increases. Both 2-D and 3-D studies were carried out in a Reynolds number (Reh) range of 5900 to 33000. They found that the reattachment length for the 3-D flow was slightly less than the 2-D case and the values of Reynolds stresses were slightly higher for 3-D flow. Moreover, the authors found an increase in turbulent intensity near the step for the 3-D case but note that this effect is negligible farther downstream of the step. Other experimental studies of significant interest that analyze vortex structures, fluctuations at wall-step and propagation of perturbations are Furuichi and Kumada (2002): Analyses of span-wise and stream-wise vortex structures; Kostas et al. (2002): Interactions of vortex structures; Lee and Sung (2002): Characteristics of wall pressure
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fluctuations; Furuichi et al. (2004): large-scale structures and fluctuations at wall step; Lee et al. (2004): Large scale vortical structures; Camussi et al. (2006): Wall pressure perturbations propagation at low Reynolds number; and Ke et al. (2005): Flow with and without entrainment and large-scale structures.
2.3 Numerical Studies on Backward Facing Steps As noted by Yang et al. (2003), the reattachment point is a critical parameter that usually determines the accuracy and performance of any numerical model. Hence, the experimental data of Armaly et al. (1983), Jovic and Driver (1994), Driver and Seegmiller (1985), Lee and Mateescu (1998) etc. are used extensively by researchers to validate their numerical studies. There are numerous 2-D and 3-D analyses of the backward facing step. This is justified, as Stephano et al. (1998) notes that every time a new numerical method is developed, it is applied and examined on backward facing step geometry to test its accuracy. A model that fails to predict the reattachment length past a backward facing step could never calculate the reattachment in complex engineering turbulent flows. This section of the review discusses the numerical studies on backward facing step geometries that include (a) 2-D and 3-D Laminar Flow Analysis, (b) Direct Numerical Simulation (DNS), (c) Large Eddy Simulation (LES) and (d) Detached Eddy Simulation (DES).
2.3.1 Laminar Flow Analyses: 2D and 3D Early numerical simulations of the step were restricted to the two-dimensional analysis primarily due to the lack of computer power. Armaly et al. (1983) conducted a
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numerical analysis and found that the results matched quite well with the experimental results up to Re ≈ 400. They hypothesized that the presence of secondary recirculation zone was the cause of the discrepancy between the experimental and numerical solutions. Other significant numerical studies of that period that discuss the 2-D numerical analysis over the step were Goussibaile et al. (1984), Toumi et al. (1984), Ecer et al. (1984) and Braza et al. (1984). It is useful to note that all the above papers use finite difference method for the numerical analysis. Kim and Moin (1985) found that their results agreed well with the experimental data of Armaly et al. (1983) until Re = 600. After that their results started to digress and the deviation was recognized as due to the three dimensionality of the flow. Kaiktsis et al. (1991) performed a 3-D analysis and noted that the primary reason for the discrepancies between the experimental and numerical studies is the onset of three dimensionalities at these Reynolds numbers. They also presented the fact that irrespective of the exactness of the numerical solution, it always under estimated the reattachment length at Re=600 (see Figure 2-9). Williams and Baker (1997) observed for the first time that due to the side walls a ‘wall- jet’ is generated from the side walls towards the mid-plane of the channel. Their two-dimensional computations under predicted the re-attachment lengths after Re ≈ 400. However, the 3-D simulations were able to correctly predict the experimental results, thus confirming the effect of three-dimensionality in the flow field after Re ≈ 400.
15
. Figure 2-9: Re-Attachment Length vs. Reynolds number for Laminar Flow from Kim and Moin (1985); xr: Re-attachment length (XR)
Chiang and Sheu (1998) conducted extensive 3-D analysis in the Reynolds number range of 100 < Re < 1000 and found that two dimensionality in the flow is achieved at Re = 800 only when the channel width is almost 100 times the step height. Their results showed excellent agreement with the experiments of Armaly et al. (1983) for Re between 100 and 389. They applied topology theory to the numerical analysis and studied the complex vortical flow structure in explicit detail. Similar experimental and numerical results were obtained by Tylii et al. (2002) Biswas et al. (2004) presented a review of previous studies and also conducted numerical investigations on backward-facing step flows with expansion ratios of 1.9423, 2.5 and 3.0. Their investigation of step flows (0.0001 < Re < 800) yielded results akin to Williams and Baker (1997) and Chiang and Sheu (1998). Moreover, the authors also
16
evaluated pressure losses for various expansion ratios and found that the losses decreased with an increase in Reynolds number and decrease in step-height. Gualtieri (2005) investigated 2-D step-flow using commercial software FEMLAB. The simulations were performed for a Reynolds number range of 84 to 1006. Their results indicated that beyond Re = 300, the re-attachment length was under predicted by FEMLAB software. Kanna and Das (2006) analyzed the backward facing step flow using the streamfunction – vorticity approach. They found that at Re= 800, even though good agreement was observed for the u-velocity after re-attachment, considerable disagreements were found in the v-velocity. The authors comment that these discrepancies contribute directly to the measurement accuracy of the re-attachment of primary vortex.
2.3.2 Direct Numerical Simulation (DNS) The Navier-Stokes (N-S) equation correctly describes both the laminar and the turbulent flows of a Newtonian fluid. One of the most powerful techniques to be developed to solve the N-S equations is undoubtedly the Direct Numerical Simulation (DNS). But unfortunately, this technique is extremely expensive and the computational cost increases as the order of Re3. Moreover, most of the effort (almost 99.8 %) is used to simulate the flow in the dissipation scales (Pope, 2000). These smaller scales can obviously be modeled, while the large scales can still be simulated. This is the basis of the technique: Large Eddy Simulation (LES). LES can save considerable amount of computational time as compared to DNS but even LES has not evolved enough to find its way as a practical engineering tool. The working engineer still relies on the traditional
17
approach of Reynolds-averaged-Navier-Stokes (RANS) equation for the solution. The various models of RANS equation i.e. the k-ε models will be discussed later in the section. One of the most comprehensive analyses of the backward facing step was the Direct Numerical Simulation (DNS) carried out by Le et al. (1997). The Reynolds number (based on step height and inlet velocity) at which the computations were done is 5100. The grids used were 768, 192 and 64 in the x, y and z directions respectively. They found that the re-attachment point varied in the span wise direction and it oscillated about a mean value of 6.28 S. Their results are in excellent agreement with the experimental data of Jovic and Driver (1994). Their extensive analysis contains up to third order statistics and Reynolds stress budgets at every location in the flow field. The DNS for the first time reported (a) the presence of a large negative skin friction in the recirculation region at relatively low Re (which agreed with the experimental readings) and (b) deviation of the velocity profile from the log law in the recovery region. This indicates that the flow is not fully recovered even at twenty step heights behind the step. Valsecchi (2005) conducted DNS for transitional flow over a backward facing step for Re = 3000 and an expansion ratio of 1.09. They found that the DNS results were in good agreement with the experimental results. Other applications of DNS to simulate a passive control method (thereby reducing the reattachment length) were studied by Neumann and Wengle (2003). They found that a certain minimum distance between the step edge and control fence (a small obstruction upstream of the step that causes the flow to be turbulent) is required to achieve maximum reduction of reattachment length. Other
18
significant studies that analyze periodically perturbed flow using LES and DNS are Dejoan et al. (2005) and Saric et al. (2005).
2.3.3 Large Eddy Simulation (LES) Silveira et al. (1991) performed Large Eddy Simulation (LES) over a backward facing step using finite-volume method. They compared the results to the experiment of Eaton and Johnston (1980) and Table II shows the comparison of results.
Table II: Comparison of Experimental and Numerical Approaches of Silveira et al. (1991) Study #
Re-attachment Length (h = step-height)
Experiment by Eaton and Johnston (1980)
7.8 h
LES by Silveira et al. (1991)
8.1 h
Grid Independent k-ε results by Silveira et al. (1991)
6.2 h
Other recent studies on Large Eddy Simulations include Inagaki et al. (2005); Benhamadouche et al. (2006); Petry and Awruch (2006) and Popiolek et al. (2006). The authors validate their new LES models on backward facing steps and comparisons to experiments are presented.
2.3.4 Turbulent Flow Analysis: Modeling of RANS The most popular method of analysis of the turbulent flows is the standard k-ε model. As noted by Yang et al. (2003), the k-ε models are equally feasible for scientific
19
research as well as for engineering applications of complex turbulent flows. The turbulent flow models can be classified as (a) linear models and (b) non-linear models. Two of the linear models are the standard k-ε model of Launder and Spalding (1974) and the non-equilibrium model of Yoshizawa and Nisizima (1993). As noted by Pope (2000), the standard k-ε model is not able to capture the secondary turbulent flows in a duct with non-circular cross-section. As a result, important non-linear models were formulated primarily to overcome this deficiency of the standard k-ε model. Quadratic models include those developed by Speziale (1987) and Shih, Zhu and Lumley (1995), while an example of a cubic model is the one developed by Craft, Launder and Suga (1996). Yang et al. (2003) compare all the linear and non-linear models mentioned above and observe that all the models under-predict the reattachment length. They also find that the non-linear models of Shih, Zhu and Lumley (1995) and Craft, Launder and Suga (1996) perform better than the linear models, and are closer to the experimental value of re-attachment length. Bredberg et al. (2002) presented an improved version of the k-ω model. Their model dispenses with the wall function and near wall information and is completely integrable through the near the wall region. The new k-ω model was compared to DNS (Le et al. (1997)) and other results over a backward facing step and the results were found to be satisfactory as shown by Figure 2-10.
20
Figure 2-10: Skin Friction Co-efficient vs. x/h from Bredberg et al. (2002)
Kim et al. (2005) simulated the experiment of Driver and Seegmiller (1985) using commercial software FLUENT. Their study confirmed that different combinations of turbulence models and wall treatment methods resulted in varying re-attachment lengths. Celik and Li (2005) presented a numerical uncertainty analyses on turbulent flow simulations over a backward facing step using FLUENT software. The authors concluded that four sets of carefully selected grids were adequate for uncertainty analysis and grid convergence.
2.4 Experimental Studies on Porous Media In this section of the literature review, recent advances in turbulent flow modeling in porous media are presented. Two distinct sets of modeling approaches can be observed from the literature depending on the order of integration i.e. starting with spaceaveraging (space-time approach) or starting with time-averaging (time-space approach).
21
These methodologies are reviewed in detail and the various models with different complexities are discussed. Moreover, conclusions are presented with regards to their accuracy and flexibility of application. The flow in porous media has been generally considered laminar due to the relatively small pore size. However, experimenters did find instances of chaotic or turbulent flow in porous media. In the literature one finds only a handful of experiments on turbulent flows in porous media. Dybbs and Edwards (1984) studied the flow of water and various oils through a fixed three dimensional packing of Plexiglas spheres and cylinders, shown in Figure 2-11. Laser anemometry and flow visualization revealed four distinct flow regimes as summarized in Table III. Here, ReP is the pore Reynolds number which is defined as the Reynolds number based on the pore size and ReP is given by Equation 2.1.
Re P =
ρU P d P µ
(2.1)
(where ρ: fluid density, µ: fluid viscosity, UP: pore velocity and dP: pore diameter) In the Darcy or Creeping flow regime, viscous forces dominate the flow and Darcy’s equation (Equation 2.2) governs the flow. In this region relationship between the pressure drop and flow rate is linear. From Re = 1 to 10, the authors observed the formation of a boundary layer on the solid surfaces of the porous media and an inertial core. The pressure drop-flow rate relation turns non-linear in this region. This continues until Re = 150 which is characterized by steady laminar flow and unsteady flow persists until Re = 300. The Reynolds number investigated in this study ranged from 0.16 to 700.
22
1 µ ∆ p = − v + C 2 ρ v 2 b 2 α
(2.2)
(where α: permeability, C2: inertial constant, v: velocity and b: filter thickness)
Table III: Flow Regimes in Porous Media Summarized from Experiment by Dybbs and Edwards (1984) Range of ReP
Regime
ReP < 1
Darcy or Creeping Flow
1< ReP < 10
Inertial Flow
1 < ReP < 150
Steady Laminar Flow
150 < ReP < 300
Unsteady Laminar Flow
ReP > 300
Turbulent Flow
Figure 2-11: Cross section of Porous Medium Packed Bed of Spheres, from Dybbs and Edwards (1984)
23
Seguin et al. (1998a) present a review of similar porous media experiments and comment that the ranges of Re for the regimes depend on the particular geometry of the packing and hence are not universal. With the help of electro-chemical microprobes inserted in various test-sections, the authors found the stable laminar regime extending to ReP=180 for various test sections. Seguin et al. (1998b) found the transition to occur from 180 < ReP < 300. Thus for ReP > 300, flow is considered to be turbulent and accurate representation of the flow is possible only by turbulence modeling. In the next section, various turbulent flow models developed for porous media are reviewed.
2.5 Turbulent Flow Modeling in Porous Media Pedras and deLemos (2001) presented a classification of turbulent flow modeling in porous media based on the order of integration i.e. space averaging or time averaging. In the next section, the same convention is followed, which we will denote as ‘Space and Time’ approach (space averaging is done before time averaging) and ‘Time and Space’ approach. However in our discussion, the models are further classified into zero, one and two equation models. It is observed that the zero and one equation models have less unknown coefficients and hence can be determined by experiments. However, the twoequation models, despite being thorough in approach, lack the validation of experiments. The next section discusses in detail the different turbulent models for flow through porous media.
24
2.5.1 Space and Time Approach 1. Two Equation k-ε Models: Lee and Howell (1987) proposed a simplified version of the standard k-ε model to analyze fluid flow in porous media. Their model was one of the first efforts to model turbulent flow in porous media and hence considers a lot of assumptions that simplifies the analysis to a great extent. Forchheimer’s resistance term is considered in their modified Darcy’s equation but they do not include the Brinkman’s term that reflects the viscous diffusion effects (See Equation 3.8). The momentum equations are considered for both the homogenous fluid flow and the flow through porous media. The partial differential equations for k (turbulent kinetic energy) and ε (turbulent dissipation) are derived from the momentum equations by assuming that the absolute value of the velocity does not depend on the turbulent flow. The NavierStokes equations are not time-averaged and hence the model doesn’t account for the Reynolds stresses caused by the turbulence. However, they consider the effective viscosity as an algebraic sum of molecular viscosity and an eddy viscosity. This effective viscosity is considered in the calculation of the viscous diffusion term. The value of coefficients used for modeling in porous media is the same as used for plain fluid flow. Antohe and Lage (1997) developed a k-ε model in which all the terms of the extended Darcy’s equation are present. Time averaging of Navier-Stokes equation was done to obtain momentum equations and the closure was obtained by modeling the Reynolds stresses by solving PDE’s (Partial differential equations) for k and ε. Due to the time-averaging process, their analyses are more complex than Lee and Howell (1987) and leads to extra terms from Darcy and Forchheimer’s in the equations of k and ε. Their
25
derivation results in the conclusion that the Darcy’s term damps the flow while the effect of Forchheimer’s term on k is inconclusive. They also suggest separate sets of equations (with extra terms) for application at low Reynolds numbers. It is interesting to observe that the value of the coefficients used in the equations is the same as that for pure fluid at low Reynolds numbers. This is due to the lack of experimental data for turbulent flow through porous media. They presume that these constants may be a function of porosity at lower porosities. Antohe and Lage (1997) found that for a steady one-dimensional fully developed flow, macroscopic turbulence cannot sustain in a porous medium i.e. they found that the only solution for this case is k = 0 and ε = 0. This is the main disadvantage of the model. This is attributed to the exclusion of microscopic turbulent quantities in the determination of macroscopic quantities due to the space-averaging of the equations before the timeaveraging process. This is because, during space averaging of the microscopic fluctuations of any quantity, the smoothing of macroscopic results takes place. Further time-averaging simply results in excluding the effect of these fluctuations on the macroscopic quantity. This drawback has brought forward the issue that the order of time-averaging and space-averaging may be important and is further explained in the models of Pedras and deLemos (2001). Getachew et al. (2000) presented a revised version of k-ε model proposed by Antohe and Lage (1997). The primary difference between the two approaches being the use of a second-order correlation term in the Forchheimer’s resistance term in the timeaveraged momentum equation. The main effect due to this modification stems from the fact that these terms give rise to additional coefficients that are able to reflect the
26
microscopic turbulence effects in a better manner. The authors believe that representing Forchheimer’s term as higher order correlation terms will avoid excluding vital effects of turbulent flow in porous media. As discussed above, these additional terms appear in the differential equations for k as third-order moments (or triple velocity correlations). These are modeled by using techniques similar to Hanjalic and Launder (1992) (cited from Getachew et al. (2000)).This model is more precise than the one by Antohe and Lage (1997) for the case when the kinetic energy is much greater than the dissipation. The dissipation equation also turns out have additional unknown double and triple moments of various quantities. In general, the authors choose to model these unknowns in terms of mean velocity gradients, Reynolds stresses and dissipation. The authors also note that there are several new undetermined coefficients that can be found only by experiments on turbulent flow in porous media. Similar to the procedure adopted by previous investigators of turbulent flow modeling in porous media, the present authors are forced to adopt the coefficients for the limiting case of α→ ∞ and φ = 1, which are the conditions for pure homogenous fluid flow.
2. RNG model: Avramenko and Kuznetsov (2006) proposed a RNG (renormalization group) model for flow through porous media. Their methodology is similar to that of Antohe and Lage (1997), in the sense that their model does not reflect the microscopic fluctuations in the macroscopic equations. However, they use the RNG approach (developed by Yakhot and Orszag (1986)) while Antohe and Lage (1997) use the time-averaging procedure. Their results confirm that the porous media decreases the
27
velocity and flattens the velocity profile. The closure is obtained by using the mixing length model of turbulence.
2.5.2 Time and Space Approach 1.
Zero Equation Models: Masuoka and Takatsu (1996) proposed a zero-
equation turbulence model for porous media. The porous media considered is in the form of a packed bed of solid spheres. The microscopic flow equations in the porous media were averaged over a control volume to arrive at the governing macroscopic equations. The eddy viscosity has been modeled in such a manner that it reflects the complex microscopic turbulent diffusion process and the geometry of the porous media. They assume that the Forchheimer’s term primarily arises due to the turbulent diffusion in porous media. The eddy viscosity is assumed as a sum of pseudo-eddy viscosity and void-eddy viscosity. The pseudo-eddy viscosity (of the order of diameter of the sphere) is thought as a long distance momentum transport that physically represents the forced flow distortion while the void-eddy viscosity (of the order of α ; α is the permeability) reflects the short-distance momentum transfer and affects the shear flow over the solid spheres, as shown from the schematic in Figure 2-12. Masuoka and Takatsu (1996) note that the equation modeled is very similar to the empirically derived Forchheimer’s term, which confirms its relationship with turbulent diffusion and the void vortex. They attribute that the Forchheimer’s term physically represents the diffusion of void vortices while the thermal dispersion represents the effect of diffusion of pseudo vortices. Masuoka and Takatsu (1998), performed flow visualization experiments by dye emissions and found evidence of flow dispersion and diffusion at high Reynolds
28
numbers. Masuoka et al. (2002) further validated their model by PIV (Particle Imaging Velocimetry) visualizations and found that the transition in porous media occurs at Re > 300. Masuoka and Takatsu (2005) used LIF (Laser Induced Fluorescence) and PIV to obtain flow visualizations at various Reynolds numbers which confirmed that the phenomena of production and dissipation are intrinsic to the porous media. They attribute the production to be caused by large vorticities due to wall effects and dissipation of large vortices into smaller ones by the solid matrix.
Figure 2-12: Schematic of Pseudo and void vortices from Masuoka and Takatsu (1996)
2.
One Equation model: Alvarez et al. (2003) developed a one equation
turbulence model in which the dissipation is assumed to be function of turbulent kinetic energy and velocity. Thus the only partial differential equation is for k, where the production term is assumed to be proportional to the cube of velocity. This results in the reduction of number of unknown coefficients to just four. These are further determined 29
by experimental investigation of airflow through PVC spheres. This semi-empirical one equation model compares well for experimental fluid flow and heat transfer results primarily due to the values obtained for the coefficients.
3.
Two Equation Models: The two equation models have been the most
popular approach in the literature owing to their capability to model intricate features of the turbulent flows. In order to find the value of undetermined coefficient (Cε) that appears in the differential equation for ε, Chung et al. (2003) performed experiments on a channel with micro-tubes. Microscopic experimental analysis of the fluid flow (Friction Factor) and heat transfer (Nusselt Number) were conducted and the authors found that for a Cε value of 0.99, the friction factors and Nusselt numbers obtained numerically matched well with the experimental results. A modified form of the equations is presented which overcome the numerical deficiency highlighted by the model of Antohe and Lage (1997). This shortcoming is overcome by representing the source term as two different terms, one each for homogenous flow and flow in porous media. For conditions of high permeability and high porosity (α→ ∞ and φ = 1) i.e. for homogeneous fluid, the porous media source term automatically vanishes and a mathematical solution is obtained. The authors go on to prove that the value of the constant is independent of the Reynolds numbers and porosity. However, the authors have assumed local thermal equilibrium conditions and hence the model cannot be extended to inlet regions and nonuniform porous media. Pedras and deLemos (2001) developed a two-equation turbulence model in which the undetermined constant is proposed by numerical solution of structured porous media
30
in the form of an infinite array of circular rods. The authors confirm that the solution of macroscopic momentum equations is independent of the order of integration. (Space averaging and time averaging). However, the turbulent kinetic energy depends considerably on the order of integration. In the present analysis, local time average is considered first and then the space-average is taken. This approach is the reverse of the methodologies of Antohe and Lage (1997) and Lee and Howell (1987). The value of the constant was found to be 0.28 and the macroscopic data for turbulent kinetic energy and dissipation matched well with the values from Nakayama and Kuwahara (1999). The slight difference between the two data is attributed to the fact that Nakayama and Kuwahara (1999) used an array of square cross-section rods. Pedras and deLemos (2003) tested the model coefficient on arrays of transversely and longitudinally elliptical rods and found that it is reasonable to assume the same value for the constant in the k-equation irrespective of the porous media geometry. Their results were in close agreement to circular and square cross sections results from the literature. Chandesris et al. (2006) developed a macroscopic turbulence model in which the core of a nuclear reactor is modeled as a porous media due to similarities in the two geometries. Thus, this model serves to simulate flows in pipes, channels and rodbundles. The unknown coefficients that arise from the modeling expressions are obtained from microscopic equations which are validated against the existing experimental and DNS (Direct Numerical Simulation) data for channel flows, pipe flows, arrays of square rods and circular rods. The authors use the same model for macroscopic turbulent viscosity as used by Pedras and deLemos (2001) and Nakayama and Kuwahara (1999).
31
4.
Large Eddy Simulation (LES) and FLUENT: Gullbrand and Wirtz (2005)
computed the flow field in a lattice of mutually perpendicular cylinder, numerically by LES and compared the results of Spalart-Allmaras and k-ω models available in FLUENT. They compared the friction factors, kinetic energy and shear stresses obtained from FLUENT models to LES results. Large over and under-predictions were observed for the different models and no conclusive pattern was observed.
2.6 Backward Facing Step and Porous Media Flow over a backward facing step has been a classical test case for testing numerical models and for experiments to get insight into complex phenomena such as separation and recirculation. With porous inserts downstream of the step, the problem becomes particularly complex and finds its application in the design of air-filter housings, heat exchangers, electronic packaging etc. This section reviews the research conducted in this area. Chan and Lien (2005) analyzed the flow over backward facing step based on the approach of Lee and Howell (1987). The porous insert is placed exactly at the step and the flow downstream is studied by changing permeability, Forchheimer’s constant and the thickness of the porous insert. All the three parameters result in reduction of the re-circulation region. The effect can be observed from Figures 2-13, 2-14 and 2-15.
32
Figure 2-13: Sensitivity of flow field (stream-traces) to changes in Darcy number for b/h = 0:3 and F = 0:55 from Chan and Lien (2005)
Figure 2-14: Sensitivity of flow field (stream-traces) to changes in the Forchheimer’s constant for b/h = 0:3 and Da = 0:01 from Chan and Lien (2005)
33
Figure 2-15: Sensitivity of flow field to changes in the thickness of porous insert for Da = 0:01 and F = 0:1 from Chan and Lien (2005)
Assato et al. (2005) applied various linear and non-linear k-ε models on flow over a backward facing step with a porous medium. They found that non-linear models were better in the prediction of reattachment length even though both types of models underpredict the experimental value of reattachment length. The effect of thickness of porous media was found to be more pronounced on the flow than porosity and permeability. They found that an increase in porous media thickness caused the recirculation to disappear. See Figures 2-16, 2-17 and 2-18.
34
Figure 2-16: Comparison of streamlines between the linear and nonlinear models for backward-facing-step flow with porous insert, α = 10–6 m2, φ = 0.65 from Assato et al. (2005)
Figure 2-17: Comparison of streamlines between the linear and nonlinear models for backward-facing-step flow with porous insert, α = 10–6 m2, φ = 0.85 from Assato et al. (2005)
35
Figure 2-18: Comparison of streamlines between the linear and nonlinear models for backward-facing-step flow with porous insert, α = 10–7 m2, φ = 0.85 from Assato et al. (2005)
2.7 Previous Work at OSU As noted by Yao (2000), most researches in the field of porous media investigate the heat and mass transfers of flow fields interacting with porous media. Yao (2000) also notes that the main features of the flow inside actual air filter housing are that (a) the mean flow and the filter surface are not perpendicular and (b) the separated flow occupies a large portion of the flow domain due to the sudden expansion at the inlet. Yao (2000) was the first work to study the flow that impinges into the air filter, with the filter located in the separation region. Earlier, Newman et al. (1997) had conducted experiments on various air filter housings that were very different from the real vehicle housings. The velocity profiles upstream of the filters were measured by Laser Doppler Anemometer (LDA) and they found that filter efficiency was predicted to change with differences in velocity distributions. Later, Al-Sarkhi et al. (1997) measured velocity profiles (upstream of the
36
filter) in housings that were similar in shape to real air-filter housings. Their experimental results indicated that the filter performance can be enhanced by a uniform flow impinging normally into the filter. They also found that the mean velocity distributions are much flatter for flows with filters, due to the resistance offered by the filter. Also, over the previous decades, the step flow has been a classic test case for many experiments and numerical simulations. Hence, a large amount of experimental and numerical data is available, so that the results of our simulations can be compared to these values and can be validated. Yao (2000) conducted detailed analyses of a two-dimensional step flow, with and without the filter. For the no filter laminar case, the numerical results for the reattachment length match with the experimental results of Armaly et al. (1983) up to Reynolds number of 650. He found that the when the porous medium is placed at a location far downstream from the step, it does not affect the separated flow and the results are almost similar to the no-filter case. However, the porous medium forces the flow to redistribute i.e. the velocity in the centre decreases and the velocity near the walls increases. When the porous medium is placed in a location where the non-porous flow is separated at one side and not separated at the other, he found that the porous medium caused the flow to reattach at one side and to separate at the other side. Moreover, when the medium is placed very close to the step, he noted that the separated flow does not penetrate into the medium. It always re-attached upstream of the porous medium. But, the secondary re-circulation region was pushed upstream towards the inlet. Experiments conducted by Yao (2000) were at higher Reynolds number and not in the laminar regime. This is because; they encountered non-uniform distribution in the
37
seeding particles during LDA measurements in the low Reynolds number experiments. Measurements were taken at four Reynolds number between 2000 and 10000. Similar characteristics of re-attachment point were observed in the turbulent flow regime as well. The re-attachment point in the turbulent flow was independent of the Reynolds number and more a function of the expansion ratio. Yao (2000) performed LES at various Reynolds numbers and found that the LES was under predicting the re-attachment length. However, the LES results obtained by Yao (2000) were not definitive.
2.8 Conclusions of the Review The review finds that the only numerical analyses that predict the experimental reattachment length with good accuracy are the laminar flow analyses (for Re < 600); the Direct Numerical Simulations (DNS) and Large Eddy Simulations (LES) to some extent. Very few numerical studies in the literature include Reynolds number range of 2000 to 10000. Experimentation in porous media has been scarce in literature due to the complex structure of the solid porous matrix. Hence majority of studies in this field (porous media) are numerical in nature. Moreover, commercial software FLUENT has been used very rarely in literature to validate experimental studies. Hence, the present study aims to study the effect of porous media on the re-circulation region formed by the flow over backward-step by using FLUENT. Through this study, one can examine the performance of FLUENT in validating macroscopic experiments on porous media.
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CHAPTER 3
NUMERICAL APPROACH
3.1 Introduction Simulation of turbulent flows is an extremely challenging problem that has perplexed researchers for a good part of this century. Turbulent flows are characterized by unsteady and non-periodic motion. This results in the fluid properties exhibiting random 3 – D variations. Moreover, there is strong dependence of flow on initial conditions. The problem is further complicated by the presence of wide range of scales that require extremely fine grids (if all the scales are resolved). The most advanced computational tool available to simulate flows is undoubtedly the Direct Numerical Simulation (DNS). As discussed in Chapter 2, DNS is limited in its application for only low Reynolds number flows and simple geometries. Moreover, time and space details obtained from DNS are not required for the present problem of engineering design of airfilter housings. Time-averaged quantities are suitable for most engineering design applications. Large Eddy Simulation (LES) offers some distinct advantages over DNS. In LES, the computational domain is divided into two distinct scales: (a) large scales, where Navier-Stokes equations are directly computed and (b) small scales, which are modeled (as opposed to DNS). As seen from the literature LES has been proven to work well for
39
moderately high Reynolds numbers. The other alternative for flow simulations is the Reynolds Averaged Navier Stokes (RANS) approach. Decomposing the velocity in terms of mean velocity and fluctuating velocity (Equation 3.1) followed by time averaging of the Exact Navier-Stokes equation results in the RANS equation (Equation 3.2) u i = u i + u i′
( )
(3.1)
(
)
∂p ∂ ∂ ∂ + ρ ui + ρ ui u j = − ∂t ∂x j ∂ xi ∂ x j +
(
∂u µ i + ∂u j − 2 δ ∂u i ∂ x j ∂ x j 3 i j ∂ xi
∂ − ρ u i′u ′j ∂x j
)
(3.2)
The last term on the right hand side of Equation 3.2: − ρ ui′u ′j are the Reynolds stresses that need to be modeled. The different approaches of modeling the Reynolds stresses and the various turbulence models are discussed in detail in Section 3.3. FLUENT offers the researcher the option of modeling by applying the various turbulence models including (a) Spalart – Allmaras (b) Standard k-ε (c) RNG k-ε (d) Realizable k-ε (e) Standard k-ω and (f) Reynolds Stress model. GAMBIT is the pre-processing software for FLUENT used for creating and meshing 2-D and 3-D models. GAMBIT was used in the present study for (a) the creation of geometries of Armaly et al. (1983) and Yao (2000), (b) grid generation and (c) meshing of the geometries.
40
Figure 3-1: The CFD Simulation Pipeline for Fluent Preprocessing-2006 (Fluent Inc.)
For CFD simulations in FLUENT, solid-modeling can also be performed using popular commercial softwares like CATIA, Pro/ENGINEER, Unigraphics NX or SolidWorks. However, owing to the relatively simple 2-D geometries of Yao (2000) and Armaly et al. (1983), GAMBIT was preferred for modeling purposes. A general strategy for performing CFD simulations in FLUENT using GAMBIT as a pre-processor is shown in Figure 3-1. CFD Simulations are then performed in FLUENT. Finally, the analyses of CFD simulations are carried out by using the post-processing tools available in FLUENT. The next section presents and discusses the governing equations of fluid dynamics in the clear fluid region as well as in the porous media.
3.2 Governing Equations 3.2.1 Clear Fluid Region For the Cartesian 2-D clear fluid region, the general continuity equation (massconservation) is given as Equation 3.3.
41
∂ρ + ∇ ⋅ (ρ v) = 0 ∂t
(3.3)
In the above equation: ρ is the density of air and v is velocity.
( )
∂ ρ v + ∇ ⋅ ( ρ v v ) = −∇ p + ∇ ⋅ (τ ) + ρ g + F ∂t
(3.4)
The general form of Newton’s second law (momentum conservation) for the clear fluid region is given by Equation 3.4. In the above equation, τ is the stress tensor; p is the pressure drop; ρ g is the gravitational force and F is the external body force. For our problem, the effects of gravity and external body forces are neglected. The stress tensor is given by Equation 3.5; where in µ is the dynamic viscosity of air.
T 2 τ = µ ∇ v + ∇ v − ∇ ⋅ v I 3
(3.5)
3.2.2 Porous Region The fundamentals of flow through porous media are given by Darcy’s equation (1856). Darcy derived the formula for fluid velocity (v) in terms of the pressure gradient (∆p) across a porous medium (of length b) and the viscosity of the fluid (µ). The equation derived by Darcy was purely empirical, given as Equation 3.6.
42
v = −
α ∆P µ b
(3.6)
In the above equation, the constant of proportionality α is called the permeability of the porous medium. Whitaker (1986) derived Darcy’s equation from the NavierStokes equation by the method of volume averaging (See Equation 3.4). The standard volume averaged continuity (mass conservation) equation for flow through porous media is given by Equation 3.7.
∂ (φρ ) + ∇ ⋅ (φρ v ) = 0 ∂t
(3.7)
In the above equation, φ is the porosity of the filter. Porosity for any medium is defined as ratio of the volume of the fluid to the total volume. Equation 3.8 gives the volume averaged momentum equation for flow through porous media.
1 ∂(φρ v) µ + ∇ ⋅ (φ ρv v ) = −φ∇p + ∇ ⋅ (φ τ ) − v + ρ C v v ∂t α 2 2
(3.8)
The above Equations 3.7 and 3.8 assume isotropic porosity and are valid only for single phase flow. The terms −
1 µ v and ρ C v v represent the viscous and inertial α 2 2
forces due to pore walls on the fluid.
43
3.2.3 Boundary Condition at the Interface of Clear Fluid and Porous Media FLUENT’s documentation does not elucidate on how the software treats the boundary condition between the clear fluid and the porous media. The literature talks in detail regarding various jump conditions namely: stress jump, mass jump, and pressure jump boundary conditions. However, most of the applications considered were for the tangential flows rather than normal flows. Kuznetsov (1996) showed that accounting for a jump in shear-stress at the interface between clear fluid region and the porous media essentially influences the velocity profiles. Kuznetsov (1996) used the stress-jump boundary conditions suggested by Ochoa-Tapia and Whitaker (1995).
3.3 Turbulence Models in FLUENT The following turbulence models are available in FLUENT: the one-equation Spalart-Allmaras model, various versions of two-equation models and multi-equation Reynolds Stress model. All the models except the Reynolds Stress Models are based on the Boussinesq approach. The Boussinesq approach assumes that the turbulent eddy viscosity is an isotropic quantity which is not precisely true for the majority of the practical cases. Brief discussions on the various models are presented in this section. They are Spalart-Allmaras model (SA), k-ε models [Standard k-ε model (SKE); Renormalization group k-ε model (RNG); and Realizable k-ε model (RKE)], k-ω models [Standard k-ω model (SKW) and Shear Stress Transport k-ω model (SST)] and the Reynolds Stress Model (RSM).
44
3.3.1 Spalart – Allmaras (SA) The Spalart-Allmaras model is a one equation model based on the Boussinesq approach. It was developed specifically for aerospace applications that involve wallbounded flows. It has been observed that the model performs well for boundary layer flows with adverse pressure gradients. FLUENT’s User Manual notes that the SA model is relatively new and has not been validated for all types of engineering flows. To obtain closure, an additional transport equation for modified turbulent viscosity (ν ) is solved.
3.3.2 Standard k-ε (SKE) The Standard k-ε model is the most commonly used two-equation turbulent models and was developed by Jones and Launder (1972). The model solves two additional differential equations for kinetic energy (k) and dissipation (ε). It is a semiempirical model derived from phenomenological analysis and empirical results. The main advantages of the SKE model are (a) robustness in convergence, (b) cost of computation and (c) accuracy and applicability for a wide range of complex industrial flows. The two advanced versions of SKE i.e., RNG and RKE are explained in the following two sections. These versions have shown to perform better when the flow-field exhibits vortices, rotation and strong streamline curvature.
3.3.3 Renormalization Group k-ε (RNG) The RNG model uses a meticulous statistical technique developed by Yakhot and Orszag (1986). It differs from the SKE model in the following ways:
45
•
Additional term in the ε-equation for improved efficiency in case of rapidly strained flows.
•
Includes the effect of swirl on turbulence, thereby performing better for swirling flows.
•
Provides an analytical formula for turbulent Prandtl numbers.
•
Accounts for low Reynolds number effects
Thus the RNG model can be applied to a wider range of industrial flows with increased accuracy.
3.3.4 Realizable k-ε (RKE) Realizable k-ε model is more accurate than SKE and RNG models for cases where flow exhibits separation, recirculation, strong pressure gradients etc. FLUENT user manual defines the term ‘realizable’ as a “model that satisfies some mathematical constraints on the Reynolds stresses that are consistent with the physics of turbulent flows”. It differs from SKE in the following manner:
•
New formulation of turbulent viscosity ( µ t ).
•
New transport equation for dissipation (ε), derived from the transport equation for mean-square vorticity fluctuation.
3.3.5 Standard k-ω (SKW) The Standard k-ω model is a two-equation turbulence model by Wilcox (1988). Here, the author refers to ω (specific dissipation rate) as the ratio of ε to k i.e. rate of dissipation per unit kinetic energy. The SKW model can be applied to both wall-
46
bounded flows and shear flows. The model also incorporates specific changes that include low Reynolds number effects.
3.3.6 Shear Stress Transport k-ω (SST) The SST k-ω model was developed by Menter (1994). In this model, the SKW model is applied in the near wall region whereas the SKE model is incorporated in the free-stream region of the flow. This is done by converting the SKE model to a SKW model in the near wall region. A blending function is multiplied to both the models which are then added together. The function has a value of ‘1’ near the wall thereby activating SKW model and a value of ‘0’ far away from the wall which triggers the SKE model. These attributes make the SST model more accurate and reliable than the SKW model.
3.3.7 Reynolds Stress Model (RSM) The Reynolds Stress models are the most comprehensive turbulence model option available in FLUENT. To obtain the closure of RANS equations, transport equation for Reynolds stresses (from the stress tensor; Equation 3.5) along with dissipation are solved. Hence, the assumption of isotropic eddy viscosity is not utilized in the RSM models. The RSM models have the capability to predict complex flow with greater accuracy. However, FLUENT user manual notes that there may be cases where the simpler turbulence models might capture the physics of the problem better than the RSM model.
47
3.4 Grid Generation in GAMBIT Vertices were created by entering their co-ordinates. The units are not important while creating the grid in GAMBIT, but the simulation in FLUENT always requires the input parameters to be in S.I. units. For 2-D geometries the z – coordinate is assigned a default value of zero. Moreover, sufficient distances of 28 h were created downstream of the step to allow the flow to approach the fully developed condition. Edges were then created using correct pairs of vertices, followed by faces which were similarly constructed by grouping together appropriate edges. The geometry was then checked for corrections and any errors in solid modeling were edited at this stage. ‘Extracting Flow Volumes or faces (2-D case)’ basically refers to the exercise of defining the flow path through or around the different solid objects. The volumes or faces (2-D case) are then meshed in GAMBIT. The number of faces in any geometry is often the choice of modeler. However, it was observed that finer grids were better suited to geometries with more faces than those with a single face. Once the geometry is created, 1–D meshes are created by meshing the edges and finally, 2–D meshes are then generated by further meshing of the faces. Both Structured and Unstructured meshes are available in GAMBIT. In the present study, structured meshing was employed. Boundary conditions types for inlet, outlet and channel walls were specified and the mesh-file is saved in a binary format for FLUENT to recognize. Figure 3-2 shows the different edges where boundary conditions are applied in GAMBIT.
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Velocity Inlet
Porous Jump
Outflow
Wall
Figure 3-2: Boundary Conditions of the Edges in GAMBIT
3.4.1 Turbulent Boundary Layer The structure of the turbulent wall layer is discussed in brief in this section. The construction of the grids in GAMBIT depends on the near-wall modeling approach chosen in FLUENT. Solid walls are the primary sources of vorticity and turbulence. Moreover, flow separation and re-attachment depend on accurate prediction of development of turbulence near the walls. The different modeling options are discussed later in Sections 3.4.2 and 3.4.3. Hence, it is important to understand the turbulent wall layer profile (shown in Figure 3-3). Equations 3.9 and 3.10 defining the profile are in formulated in terms of dimensionless velocity (u+) and wall units (y+).
49
u+ E
Equation 3.12
Equation 3.11
A: Viscous Sub-Layer B: Buffer Layer C: Log-Law Region D: Outer Layer E: Inner Layer
D C B
A y+ = 5
ln y+
Figure 3-3: Turbulent Boundary Layer Profile in the Near-wall Region
Where,
u+ =
U uτ
y+ =
uτ yP
(3.9)
(3.10)
ν
uτ is the friction velocity defined as the square root of the ratio of wall
shear stress and density of the fluid. Region A is the viscous sub-layer (y+ < 5) where the Reynolds stress is negligible compared with viscous stresses. Equation 3.11 defines this region.
u+ = y+
(3.11)
50
Region B is the Buffer layer (or Blending region) that is characterized by (5 < y+ < 30) and lies between the sub-layer and the log-law region. The log-law region (Region C) is characterized by relation shown in Equation 3.12. (Also, see Figure 3-3)
u+ =
1
κ
ln y + + B
(3.12)
(where, the coefficients κ = 0.41 and B = 5.2) In the inner layer (yP/s < 0.1), the mean velocity is determined by the friction velocity and wall units alone, where as in the outer layer (y+ > 50) the effects of viscosity are negligible. The next two sections discuss the various near wall modeling approaches available in FLUENT and their influence on the geometry of the grids in GAMBIT.
3.4.2 Wall Function Approach The wall function approach is used for high Reynolds number k-ε models. In this approach the laminar sub-layer is not resolved. The first grid point from the wall is assumed to be in the logarithmic layer (y+ > 11). The determination of the distance of the first grid point from the wall is discussed in Section 3.4.5.
3.4.3 Damping Function Approach The damping function approach is used for low Reynolds numbers k-ε models. In this approach, the equations are integrated to the wall without assuming the universal law for the velocity profile. The damping functions correct the behavior of eddy viscosity by introducing various constants and functions. The classic model is the one by Launder 51
and Sharma (1972). It has been observed from the literature that the model predicts incorrect results (when compared to DNS and experimental results) for k and ε, especially near the solid walls.
3.4.4 Two Layer Model Approach In this approach the grid is separated into two different regions: (a) near wall region, where the effects of viscosity are taken into account and typically the k-ω model is applied (b) a fully turbulent outer region where the standard k-ε is applied. A region of y+ ≈ 30 is chosen by FLUENT to distinguish the two layers.
3.4.5 Determination of Distance of First Grid Point from the Wall The distance (yP) can be obtained from Equation 3.10 which is re-stated below as Equation 3.13.
y+µ yP = ρ uτ
(3.13)
Hence, in order to determine y, one needs to obtain the value of uτ . This is done by using the definition of Reynolds number based on friction velocity, as shown in Equation 3.14. Moreover, Reτ can be read directly from Figure 3-4, which shows Reτ as a function of Re (based on channel height).
52
Reτ =
ρ uτ s µ
(3.14)
Figure 3-4: Reτ as a Function of Reynolds Number from Pope (2000)
The next two sections present the a few samples of final grids used for simulating flow through the geometries of Armaly et al. (1983) and Yao (2000).
3.4.6 Samples of Final Grids Figures 3-5, 3-6 and 3-7 show some sample grids generated for both Armaly et al. (1983) and Yao (2000). Figures 3-5 and 3-6 employ structured grids and are typically used for Spalart-Allmaras, k-ε models and Reynolds Stress Models. Figure 3-7 shows a clustered grid used for k-ω models.
53
Figure 3-5: Sample Structured Grid of Armaly et al. (1983)
Figure 3-6: Sample Structured Grid of Yao (2000) – no-filter case
Figure 3-7: Sample Clustered Grid of Yao (2000) – filter at 4.25 step heights
54
3.5 Simulation in FLUENT The user interface of FLUENT is written in a language called ‘Scheme’. Scheme is a dialect of the software language LISP. FLUENT allows the user to write menu macros and functions, thus enabling the advanced user to customize the interface. The menu driven interface of FLUENT is run using a UNIX (Sun Solaris) platform. The following modeling steps are performed for running simulations in FLUENT: 1. Reading the grid file (in .msh format): File → Read → Case… The grid that is already constructed in GAMBIT is saved as a (.msh) file. This file is imported and read into FLUENT. 2. Checking the grid: Grid→ Check… This is done to ensure that there are no ‘negative cell volumes’ in the grid. Negative cell volumes are indications of improper connectivity in the grid that must be avoided. 3. Specifying solver properties: Define → Models →Solver… Options are available in FLUENT for choosing (a) Segregated or (b) Coupled solvers. In segregated solvers the continuity and momentum equations are solved separately while in coupled solvers the equations are solved simultaneously. Figure 4 shows default options given by FLUENT. 4. Selecting the physical models: Define → Models → Viscous… Depending on whether the flow is laminar or turbulent, the physical models are chosen. The default values of constants provided by FLUENT can also changed by the user.
55
5. Specifying fluid properties: Define → Material Properties… Default values of density and viscosity of air are taken from FLUENT and Table IV show the respective values.
Table IV: Physical Properties of Air at 20 oC Property
Value
SI units
Density (ρ)
1.225
kg/m3
Viscosity (µ)
1.7894 * 10-5
N·s/m
6. Specifying Boundary Conditions: Define → Boundary Conditions… The boundary conditions for the different edges can be created while constructing the geometry of the grid in GAMBIT. However, FLUENT also provides the option of modifying them using this option. Table V shows the boundary conditions used for the present geometry (See Figure 3-2 for representation).
Table V: Boundary Conditions for Backward Facing Step Geometry Region
Boundary Condition used
Inlet
Velocity Inlet
Outlet
Outflow
Channel top and bottom walls, Step wall
Wall
Filter face
Porous jump
Other cross sections in the channel
Interior
56
The Velocity Inlet boundary condition specifies the inlet velocity (in m/s) and requires the specification of a length scale such as the ‘Hydraulic diameter’ of the inlet channel and the turbulent intensity. FLUENT provides a default value of 10% for turbulent intensity. The inlet velocities are calculated by Equation 3.15 and the values for different Reynolds numbers are shown in Table VI. In Equation 3.15, Dh has a value of 0.05 m (i.e. twice the size of the inlet channel height, 25 mm).
Re =
ρ U bulk Dh ρ (1.024 × U inlet ) Dh = µ µ
(3.15)
Table VI: Input Values for Velocity Inlet Boundary Condition Case #
Reynolds Number (Re)
Input values for X-velocities (m/s)
1
2000
0.5706
2
3750
1.07
3
6550
1.8687
4
10000
2.853
The Outflow boundary condition specifies fully developed flow characterized by no further changes in horizontal and vertical velocities with respect to x direction. The Wall boundary condition implies the no-slip condition while the Porous Jump Boundary Conditions are used for the filter face. FLUENT provides two options for modeling flow through porous media. They are (a) Porous Media boundary condition and (b) Porous jump boundary condition. The latter is a 1-D simplification of the former and recommended for
57
modeling flows through filters, screens etc. Especially for cases that are not concerned with heat transfer. The porous jump model is more robust and yields better convergence. However, in the present study ‘Porous Media’ option is used. The relation between pressure and velocity used in FLUENT is given in Equation 3.16.
1 µ ∆ p = − v + C 2 ρ v 2 b 2 α
(3.16)
User inputs for permeability (α), pressure jump coefficient (C2) and thickness of the media (b) are provided through the pressure-jump panel. Tebutt (1995) obtained the values of permeability and pressure-jump coefficient by performing experiments on a single filter sheet of 1 mm thickness. Yao (2000) used a pleated filter in his experiments. The values of parameters used by Yao (2000) in his LES studies are shown in Table VII.
Table VII: Input Values for Porous Jump Boundary Conditions Property
7.
Value
SI units -9
m2
Permeability (α)
1.17 * 10
Thickness (b)
15 * 10-3
m
Pressure Jump Coefficient (C2)
4.53 * 103
1/m
Adjusting the solution controls: Solve → Controls → Solution … The momentum and the transport equations (for k and ε) are discretized using second-order upwind method. The input values for under-relaxation factors
58
are crucial since they assist in achieving convergence. However, reduction of under-relaxation factors results in longer time to reach convergence. 8.
Initializing the iterations: Solve → Initialize … The simulation is initialized by specifying the inlet velocity. FLUENT then calculates the initial values and kinetic energy and dissipation using the inlet velocity and channel height (which were previously entered).
9. Specifying Residuals: Solve → Controls …Residuals The user has an option of setting the convergence criteria. In the present study, a value of 1e-6 is used. By clicking the plot option, a graph of converging residuals can be obtained. 10. Post-Processing of solution Finally, post processing of the simulation can be done by availing the ‘Display, Plot and Report’ pull down menus. This includes vector plots, contour plots, x-y plots and other graphical options.
59
CHAPTER 4 RESULTS AND DISCUSSION
4.1 Grid Independence Studies It is extremely important in Computational Fluid Dynamics for the simulation to represent correctly the conceptual model. Moreover, the simulation should resemble real- life flows to the greatest accuracy possible. Numerical Simulations have various advantages over experiments. The primary one being that parameters can easily be changed and quick results are possible at lower costs. The details of the numerical methods and FLUENT software employed can be found from Chapter 3. In this section, grid independence studies are carried out for the geometries of Armaly et al. (1983) and Yao (2000). Moreover, the turbulent model that most closely resembles the real-world flow will be chosen for simulating two-dimensional step flows with and without the filter. Re-attachment length is used as a criterion for comparing different turbulent models and thus ensuring that the simulated flow closely resembles true experimental results. The phenomena of flow separation and re-attachment length are strongly dependent on the correct prediction of the development of turbulence near the walls. Also, excessive numerical diffusion caused by grid-density may incorrectly enhance the viscous effects leading to inaccurate simulations. This exercise enables us to observe the sensitivities of
60
the dependent variable (in our case: Re-attachment length) on multiple refined spatial grids (AIAA Editorial Policy Statement). For an accurate CFD simulation, quantifying uncertainty and error is very essential. Typically, uncertainties refer to lack of knowledge regarding modeling parameters while errors refer to a deficiency that has not been caused by lack of knowledge. In our present simulations, the assumption of the log-law in the near wall region in the re-circulation zone can be considered as an uncertainty. The main sources of error (AIAA G-077) that need to be paid attention to are insufficient spatial or temporal discretization convergence, lack of iterative convergence and other programming errors. It is generally recommended that discretization employed be at least second-order accurate. Moreover, AIAA Editorial policy states that the solution also be compared to a highly accurate numerical solution. For iterative convergence, the value of residual error (magnitude of difference between both sides of the difference equations) should be set to a very low value [Roache (2002)]. In the present study, it is set to 1e-6. The inlet velocities for both geometries are calculated from the Reynolds numbers for which the experiments are planned to be validated. For the outlet, fully developed conditions are confirmed by observation of downstream velocity profiles. The next two sections present the results of grid independence studies on the geometries of Armaly et al. (1983) and Yao et al. (2000).
4.1.1 Grid Independence: Armaly et al. (1983) Grid Independence studies were conducted at Re = 7000 for the backward-facing step geometry of Armaly et al. (1983). The experimental re-attachment length from
61
Armaly et al. (1983) was found to be around 7 h (seven step heights). The details of Armaly et al.’s (1983) geometry can be obtained from Table I. It can be observed from Table VIII that Grid #1 with an X-grid spacing of 0.5 mm under-predicts the reattachment length while Grid #3 predicts the re-attachment length closest to the experimental value. It was observed that any X-grid spacing greater than 2mm (Grid #4) resulted in further decrease of re-attachment point. The Y-grid spacing of 0.143 mm is unique for Re = 7000 and has been chosen so that the first grid point from the wall lies in the region of y+ ≈ 11 (see Section 3.4.1 for a detailed explanation). Finally, a clustered grid was constructed (according to the dimensions of Grid # 3) that utilized a finer mesh near the step and coarser mesh away from the step.
Table VIII: Various Grid Sizes Used for Realizable k-ε Model at Re = 7000 Re-attachment Length Grid #
Grid Size
1
(mm)
(step heights)
0.5 mm x 0.143 mm
29.5
6.02 h
2
1.0 mm x 0.143 mm
31.5
6.43 h
3
1.5 mm x 0.143 mm
32.25
6.58 h
4
2.0 mm x 0.143 mm
32
6.53 h
5
Clustered grid
32.25
6.58 h
The observation from Table VIII established the importance of the first grid point from the wall while conducting simulations using FLUENT i.e. the value of 0.143 mm as the Y-grid spacing for Re = 7000 was the most important factor in getting satisfactory numerical results. This study was then extended to different turbulence models in FLUENT and the results are shown in Table IX. Realizable k-ε model (with enhanced
62
wall-functions option) gave the optimum results. Even though Spalart-Allmaras model predicted the closest re-attachment length to the experimental one, it was not used for further analysis owing to the doubtfulness of its accuracy from the literature (FLUENT User’s Guide). For the k-ω models (Case 7 and 9 in Table IX, marked with *), finely clustered grids were used while for all the other models Grid #3 (from Table VIII) was used. The Reynolds stress model resulted in large simulation times along with underprediction of re-attachment length. Moreover, Kim et al. (2003) showed that k-ω models in FLUENT result in inaccurate velocity vector plots as compared to k-ε models. Hence, from the results of Table VIII and IX along with indications from the literature, the Realizable k-ε model was chosen for further simulations in this study. The next section discusses in detail the grid independence studies for the geometry of Yao (2000).
4.1.2 Grid Independence: Yao et al. (2000) When the Realizable k-ε model is used along with a Y-grid spacing of 0.75 mm for Re = 6550, excellent agreement with experimental results is obtained (see Table X). This can be also be observed from Figure 35 from Section 4.2.3. When the Reynolds number was varied (see Table XI), good agreement was observed for all Reynolds numbers except Re = 2000. Other turbulence models were tried at these Reynolds numbers along with various combinations of clustered and regular grid. However, it was found that the experimental re-attachment length of 8 h for Re = 2000, (greater than reattachment length at Re = 10000) was difficult to simulate in FLUENT. This may be attributed to the transitional flow regime from Re = 1000 to 6600 [Armaly et al. (1983)],
63
that the turbulence models in FLUENT are unable to simulate. Moreover, most of the numerical studies in literature concentrate either on the low Reynolds number (laminar regime) or high Reynolds number – DNS, LES or RANS studies. No previous studies have reported results at these Transitional Reynolds numbers. Hence, Section 4.2.1 that discusses numerical results at found unsatisfactory results at Re = 2000.
Table IX: Various Turbulence Models Used for Geometry of Armaly et al. (1983) at Re = 7000
Case #
Turbulence Model Used
Re-attachment length (h = step height)
1
Spalart-Allmaras
7.19 h
2
Standard k-ε
5.51 h
3
RNG k-ε
6.27 h
4
Realizable k-ε (Standard Wall Functions)
6h
5
Realizable k-ε (Enhanced Wall Functions)
6.58 h
6
Realizable k-ε (Non-Equilibrium Wall Functions)
5.05 h
7
Standard k-ω
5.075 h *
8
Reynolds Stress Model
6.01 h
9
SST k-ω
6h*
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Table X: Various Grid Sizes Used for Geometry of Yao (2000); Realizable k-ε Model at Re = 6550
Grid #
Re-attachment Length
Grid Size
(mm)
(step heights)
1
0.5 mm x 0.75 mm
29.5
6.13 h
2
1.5 mm x 0.75 mm
31.5
6.42 h
3
2.5 mm x 0.75 mm
32.25
6.32 h
4
3.5 mm x 0.75 mm
32
6.37 h
5
Clustered
32.25
6.32 h
Also, as discussed in Section 3.6, the porous jump boundary conditions used in FLUENT at the interface between the clear-fluid region and the porous media is not documented well in the FLUENT User’s guide. Hence, important details on how FLUENT treats this boundary condition are missing in this study. This may be one of the important factors affecting numerical results found in the present study.
Table XI: Re-attachment Lengths at Different Reynolds Numbers Using Realizable k-ε model Reynolds Number
Grid Size
2000
Re-attachment Length Experimental
FLUENT
3.5 mm x 2 mm
8
4.9 h
3750
3.5 mm x 1.15 mm
6
6.23 h
6550
3.5 mm x 0.75 mm
6.5
6.37 h
10000
3.5 mm x 0.5 mm
7
6.6 h
4.2 Numerical Results from FLUENT 4.2.1 Re = 2000 and Re = 3750 Separation lines (as seen from Figure 4-1), provide a good indication of the flow field physics along with the re-attachment point. Each point on the separation line
65
indicates the approximate position of the zero-velocity point. Figure 4-1 shows the separation lines obtained from FLUENT for the no-filter case and for filters placed at 4.25 h and 6.75 h. One can observe the drastic reduction in re-circulation region when the filter is placed at 4.25 step heights. The separation lines shown in Figure 4-1 are compared to the experimental observations of Yao (2000) in Figures 4-2, 4-3 and 4-4. They clearly indicate the inability of FLUENT to capture the physics of flow at Re = 2000. The no-filter case (Figure 4-2) and filter at 6.75 h (Figure 4-4) show appreciable difference in experimental and numerical results. These discrepancies may be attributed to the transitional flow regime from Re = 2000 to Re = 6600 [Armaly et al. (1983)] where the flow loses its two-dimensionality. From, the literature review, it can clearly be observed that none of the numerical studies have attempted to simulate flow at these transitional Reynolds numbers (Re = 2000 to 6600). The numerical studies either discuss the flow field in the extremely low laminar regimes or analyze the high Reynolds number flows using DNS, LES or RANS approach. Experimental observations of Yao et al. (2000) and Armaly et al. (1983) clearly indicate that re-attachment point at Re = 2000 for the no-filter case is much greater than even the fully turbulent Reynolds numbers (Re > 6600). As seen from Figure 4-2 the present 2-D study is clearly unable to capture to physics of the flow-field at these low Reynolds numbers. Another important observation from the studies of Yao (2000) is that separation line for filter at 6.75 h is always higher than the separation line for the no-filter case (contrary to Figure 4-1 i.e., FLUENT results). This means that a larger re-circulation region obtained when the filter is placed at 6.75 h. FLUENT is unable to capture this feature of the flow field not only at Re = 2000 but also for Re = 3750 and 6550. However, at Re = 10000, this experimental
66
observation is exhibited well by FLUENT. For the filter placed at 4.25 h (see Figure 43), an improvement in similarity between the experimental results of Yao (2000) and FLUENT is observed.
1 0.9 0.8 0.7
Y/h
0.6 0.5 0.4
No filter case
0.3
Filter at 4.25 h
0.2
Filter at 6.75 h
0.1 0 0
1
2
3
4
5
X/h Figure 4-1: Separation Lines at Re = 2000: FLUENT
67
6
1 0.9 0.8 0.7
Y/h
0.6 0.5 0.4 0.3
Experiment
0.2
FLUENT
0.1 0 0
1
2
3
4
5
6
7
8
X/h Figure 4-2: Comparison of Experiment and FLUENT: No Filter Case, Re = 2000
1 0.9 0.8 0.7
Y/h
0.6 0.5 0.4
Experiment
0.3
FLUENT
0.2 0.1 0 0
1
2
3
4
5
X/h Figure 4-3: Comparison of Experiment and FLUENT: Filter at 4.25 h, Re = 2000
68
1 0.9 0.8 0.7
Y/h
0.6 0.5 0.4 0.3
Experiment
0.2
FLUENT
0.1 0 0
1
2
3
4
5
6
7
X/h Figure 4-4: Comparison of Experiment and FLUENT: Filter at 6.75 h, Re = 2000 Similar discrepancies are observed at Re = 3750 (see Figure 4-5). However, it interesting to observe from Figures 4-6, 4-7 and 4-8 that numerical results obtained from FLUENT are closer to experimental results of Yao (2000). Figure 4-6 show that for the no-filter case, FLUENT predicts a regular separation line unlike the asymmetrical experimental curve. For the filter placed at 4.25 h, good agreement is observed especially at the end of the separation region. However, for the filter at 6.75 h, FLUENT results are unable to predict the increase in the size of re-circulation (see Figure 4-8).
69
1 0.9 0.8 0.7
Y/h
0.6 0.5 0.4
No filter case
0.3
Filter at 4.25 h
0.2
Filter at 6.75 h
0.1 0 0
1
2
3
4
5
6
7
X/h Figure 4-5: Separation Lines at Re = 3750: FLUENT
1 0.9 0.8 0.7
Y/h
0.6 0.5 0.4 Experiment
0.3
FLUENT
0.2 0.1 0 0
1
2
3
X/h
4
5
6
7
Figure 4-6: Comparison of Experiment and FLUENT: No Filter Case, Re = 3750
70
1 0.9 0.8 0.7
Y/h
0.6 0.5 0.4 Experiment
0.3
FLUENT
0.2 0.1 0 0
1
2
3
4
X/h Figure 4-7: Comparison of Experiment and FLUENT: Filter at 4.25 h, Re = 3750
1 0.9 0.8 0.7
Y/h
0.6 0.5 0.4 0.3
Experiment
0.2
FLUENT
0.1 0 0
1
2
3
4
5
6
7
X/h Figure 4-8: Comparison of Experiment and FLUENT: Filter at 6.75 h, Re = 3750
71
4.2.2 Re = 6550 and Re =10000 Yao (2000) observed experimentally that the flow fields with and without the filter at Re = 6550 and Re = 10000 are quite similar and the separation lines exhibited almost identical behavior. Similar results were observed by Armaly et al. (1983) that confirmed the onset of fully turbulent behavior at Re ≈ 6600. Hence, the present numerical computations (using FLUENT) show results closer to experimental studies of Yao (2000) at Re = 6550 and 10000.
1 0.9 0.8 0.7
Y/h
0.6 0.5 0.4
No filter case
0.3
Filter ar 4.25 h
0.2
Filter at 6.75 h
0.1 0 0
1
2
3
X/h
4
5
6
7
Figure 4-9: Separation Lines at Re = 6550: FLUENT
For the no-filter case, the re-attachment point is found close to 6.5 h. This agrees well with the experimental results, as seen from Figure 4-10. However, Armaly et al. (1983) found the re-attachment point for turbulent Reynolds numbers to be around 8h.
72
This difference as noted by Yao (2000) may be due to various factors like difference in inlet RMS velocity profiles etc. Moreover, in the present studies, no evidence of nonstep side vortex was found at Re = 6550. This observation agrees well with the experimental results of both Yao (2000) and Armaly et al. (1983). When the filter is placed at 4.25 step heights (see Figure 4-9), the re-circulation region is greatly reduced. Figure 4-11 shows that after 2h, the experimental and present results agree to some extent. The filter when placed at 4.25 h greatly alters the flow field as compared to the no-filter case. From Figure 4-13, one observes that the filter pushes the maximum velocity region towards the centre of the channel to present a symmetrical profile, while the no-filter case exhibits an unsymmetrical velocity distribution. Similar trends are observed for the velocity profiles from the studies of Yao (2000). For the filter placed at 6.75 step heights, FLUENT shows a smaller re-circulation region as contrary to experimental analyses. Moreover, a shorter re-attachment length is observed (see Figures 4-11 and 4-12). This shows that in the present numerical studies, filter placed at 6.75 h still affects the flow field while no such evidence is found experimentally. Figure 4-14 shows the comparison of velocity profiles at 5h. It can be observed that it agrees well with the trends exhibited by the no-filter case. However, after 5h, the filter case (6.75h) re-attaches quickly while no-filter case flow (experiment and FLUENT) predicts longer reattachment length. The following conclusions can be drawn for FLUENT simulations at Re = 6550. The no-filter case and filter at 4.25 h (see Figures 4-10 and 4-11) is simulated well by FLUENT. The case of the filter placed at 6.75 h still exhibits the influence of the filter on the flow field. These discrepancies may be due to the inability of FLUENT and/or the
73
realizable k-ε model that introduce excessive numerical dissipation especially near the walls. 1 0.9 0.8 0.7
Y/h
0.6 0.5 0.4 0.3
Experiment
0.2
FLUENT
0.1 0 0
1
2
3
X/h
4
5
6
7
Figure 4-10: Comparison of Experiment and FLUENT: No Filter Case, Re = 6550 1 0.9 0.8 0.7
Y/h
0.6 0.5 0.4 0.3
Experiment
0.2
FLUENT
0.1 0 0
0.5
1
1.5
2
2.5
3
3.5
4
X/h Figure 4-11: Comparison of Experiment and FLUENT: Filter at 4.25 h, Re = 6550
74
1 0.9 0.8 0.7
Y/h
0.6 0.5 0.4 0.3
Experiment
0.2
FLUENT
0.1 0 0
1
2
3
4
5
6
7
X/h Figure 4-12: Comparison of Experiment and FLUENT: Filter at 6.75 h, Re = 6550
0.06 0.05
Y (m)
0.04 No filter case
0.03
Filter at 4.25 h
0.02 0.01 0 -0.5
0
0.5
1
1.5
2
2.5
X-Velocity (m/s) Figure 4-13: FLUENT Velocity Profiles at 3.75 h; With and Without Filter at 4.25 h, Re = 6550
75
0.06
0.05
Y (m)
0.04
0.03
No filter case Filter at 6.75 h
0.02
0.01
0 -0.5
0
0.5
1
1.5
2
2.5
X Velocity (m/s) Figure 4-14: FLUENT Velocity Profiles at 5h; With and Without Filter at 6.75 h, Re = 6550
At Re = 10000, the best agreement between FLUENT and experimental results of Yao (2000) are observed. Figure 4-15 clearly shows the increase in re-circulation region when the filter is placed at 6.75 h as compared to the no-filter case. This behavior of the flow field was explicitly observed by Yao (2000) in his experiments for all the Reynolds numbers. However, the present numerical computations show this behavior only at high Reynolds numbers of Re = 10000. The no-filter case separation lines (see Figure 4-16), show the best agreement with experimental ones with both predicting a re-attachment point between 6 and 7 step heights. With the filter placed at 4.25 h, similar results to previous cases are observed i.e. reduction in re-circulation region (see Figure 4-17).
76
1 0.9 0.8 0.7
Y/h
0.6 0.5 0.4
No filter case
0.3
Filter ar 4.25 h
0.2
Filter at 6.75 h
0.1 0 0
1
2
3
4
5
6
7
X/h Figure 4-15: Separation Lines at Re = 10000: FLUENT
1 0.9 0.8 0.7
Y/h
0.6 0.5 0.4 0.3
Experiment
0.2
FLUENT
0.1 0 0
1
2
3
4
5
6
7
X/h Figure 4-16: Comparison of Experiment and FLUENT: No Filter Case, Re = 10000
77
1 0.9 0.8 0.7
Y/h
0.6 0.5 0.4 0.3
Experiment
0.2
FLUENT
0.1 0 0
0.5
1
1.5
2
2.5
3
3.5
4
X/h Figure 4-17: Comparison of Experiment and FLUENT: Filter at 4.25 h, Re = 10000
When the filter is placed at 6.75 h (see Figure 4-18), FLUENT does manage to show the trend of the flow-field exhibited by the experimental results. The re-attachment point predicted by FLUENT is around 0.5h shorter that than the experimental one. The velocity profiles at 3.75 h (see Figure 4-19), is very similar to the ones obtained for previous Reynolds numbers. However, owing to the early re-attachment of the flow for the filters placed at 6.75 h, the velocity profiles do not quite match at the 6.25 step heights (see Figure 4-20).
78
1 0.9 0.8 0.7
Y/h
0.6 0.5 0.4 0.3
Experiment
0.2
FLUENT
0.1 0 0
1
2
3
4
5
6
7
X/h Figure 4-18: Comparison of Experiment and FLUENT: Filter at 6.75 h, Re = 10000 0.06
0.05
0.04
Y (m)
No filter case Filter at 4.25 h
0.03
0.02
0.01
0 -1
-0.5
0
0.5
1
1.5
2
2.5
3
3.5
X Velocity (m/s)
Figure 4-19: FLUENT Velocity Profiles at 3.75 h: With and Without Filter at 4.25 h, Re = 10000
79
0.06
0.05
Y (m)
0.04
No filter case 0.03
Filter at 6.75 h 0.02
0.01
0 -0.5
0
0.5
1
1.5
2
2.5
3
X Velocity (m/s)
Figure 4-20: FLUENT Velocity Profiles at 6.25 h: With and Without Filter at 6.75 h, Re = 10000
4.2.3 Separation Lines Figures 4-21 and 4-22 show the overall comparison between the experimental results of Yao (2000) and FLUENT respectively for the no-filter case. One can observe that for Re = 6550 and Re = 10000, FLUENT does predict the trend of the experimental separation lines. However, as discussed earlier in this chapter, for Re = 2000 is not simulated accurately by FLUENT. The re-attachment lengths predicted for other three cases match well with the experimental results of Yao (2000).
80
1 0.9 0.8 0.7
Y/h
0.6 0.5 0.4
Re = 2000
0.3
Re = 3750
0.2
Re = 6550
0.1
Re = 10000
0 0
1
2
3
4
5
6
7
8
X/h Figure 4-21: Separation Lines for No Filter Case: Yao (2000)
1 0.9 0.8 0.7
Y/h
0.6 0.5 0.4
Re = 2000
0.3
Re = 3750
0.2
Re = 6550 Re = 10000
0.1 0
0
1
2
3
4
5
6
7
X/h Figure 4-22: Separation Lines for No Filter Case: FLUENT
81
8
Figures 4-23 and 4-24 show the separation lines for the filter placed at 4.25 h. It can be observed that the three case of Re = 2000, 3750 and 6550: numerical results match well with the experimental ones. The separation lines for the these Reynolds numbers almost coincide leading to the observation that when the filter is placed at 4.25 h, the effect of Reynolds number is not seen by the flow. FLUENT also seems to be predicting shorter re-attachment lengths for the filter placed at 4.25 h. From Figures 4-25 and 4-26, one can observe the overall comparison between Yao’s experimental studies and FLUENT for the case of filter placed at 6.75 h. The size of the re-circulation regions for different Reynolds numbers predicted by FLUENT is completely opposite (in the reverse order) when compared to experimental results of Yao (2000). From this numerical study, it can be summarized that commercial software FLUENT does exhibit the trends shown in the literature. Good agreement is observed especially for the higher Reynolds numbers flows for the no-filter case and for the filter placed at 4.25 h. Moreover, important information regarding the boundary condition at the interface between the clear-fluid and the porous media was missing in the FLUENT’s documentation. This may be one of the major factors influencing the results when the filter is placed at 6.75 h.
82
1 0.9 0.8 0.7
Y/h
0.6 0.5 0.4
Re = 2000
0.3
Re = 3750
0.2
Re = 6550 Re = 10000
0.1 0 0
1
2
X/h
3
4
5
Figure 4-23: Separation Lines for Filter at 4.25 h: Yao (2000)
1 0.9 0.8 0.7
Y/h
0.6 0.5 0.4
Re = 2000
0.3
Re = 3750 Re = 6550
0.2
Re = 10000 0.1 0 0
1
2
3
4
X/h Figure 4-24: Separation Lines for Filter at 4.25 h: FLUENT
83
5
1 0.9 0.8 0.7
Y/h
0.6 0.5 0.4
Re = 2000 0.3
Re = 3750
0.2
Re = 6550
0.1
Re = 10000
0 0
1
2
3
4
5
6
7
X/h Figure 4-25: Separation Lines for Filter at 6.75 h: Yao (2000)
1 0.9 0.8 0.7
Y/h
0.6 0.5 0.4
Re = 2000
0.3
Re = 3750 Re = 6550
0.2
Re = 10000 0.1 0 0
1
2
3
4
5
6
X/h Figure 4-26: Separation Lines for Filter at 6.75 h: FLUENT
84
7
4.2.4 Effect of Variation of Permeability, Inertial Constant and Thickness on Separation Lines For the case of the filter placed at 4.25 h at Re = 10000 (Figure 4-17), good agreement with theory is observed. Hence, this case was chosen to investigate the effects of (a) permeability, (b) inertial constant and (c) thickness of the porous media on the flow upstream of the filter. One can view from Figures 4-27, 4-28 and 4-29 that when filter is placed at 4.25 h, no changes in the flow upstream of the flow is detected by FLUENT. Moreover, similar results for variation in permeability are obtained by Yao (2000) for laminar flows when the filter is placed far downstream from the step (20.55 h). These results however are far different from the results of Chan and Lien (2005) [see Figures 213, 2-14 and 2-15] and Kuznetsov (1996). The reasons for the disparity in the present results and the literature may be due to: (a) the placement of the porous insert, which in their case was right at the step-wall, (b) the order of Darcy number, which in the present study is from 10-9 to 10-5 (as compared to 10-4 to 10-1, in the literature), (c) treatment of the boundary condition at the interface of clear-fluid and porous medium by FLUENT. However, it was observed that the velocities and the flow-field inside the filter and downstream of the filter are affected strongly due to the variation of these parameters.
85
1 0.9 0.8 0.7
Y/h
0.6 0.5 0.4
α = 1.17e-7 m2 α = 1.17e-9 m2 α = 1.17e-11 m2
0.3 0.2 0.1 0 0
0.5
1
1.5
2
2.5
3
3.5
4
X/h Figure 4-27: Effect of Variation of Permeability (α) on Separation Lines by FLUENT; Inertial Constant (C2) = 4.533*103 1/m, Thickness (b) = 15 mm for Filter Placed at 4.25 h: Re=10000
Effect of variation in Inertial constant 1 0.9 0.8 0.7
Y/h
0.6 0.5 0.4
c2 = 45.33 1/m c2 = 4533.33 1/m c2 = 4.533e5 1/m
0.3 0.2 0.1 0 0
0.5
1
1.5
2
2.5
3
3.5
4
X/h Figure 4-28: Effect of Variation of Inertial Constant (C2) on Separation Lines by FLUENT; Permeability (α) = 1.17*10-9 m2, Thickness (b) = 15 mm for Filter Placed at 4.25 h: Re=10000
86
1 0.9 0.8 0.7
Y/h
0.6 0.5 0.4
b = 3.5 mm b = 15 mm b = 25 mm
0.3 0.2 0.1 0 0
0.5
1
1.5
2
2.5
3
3.5
4
X/h Figure 4-29: Effect of Variation of Thickness (b) on Separation Lines by FLUENT; Permeability (α) = 1.17*10-9 m2, Inertial Constant (C2) = 4.533*103 1/m for Filter Placed at 4.25 h: Re=10000
87
CHAPTER 5
CONCLUSIONS AND RECOMMENDATIONS
5.1 Conclusions FLUENT software was used to simulate experiments of Yao (2000) for no-filter case and for filters placed at 4.25 and 6.75 step heights. Reynolds number was varied from Re = 2000 to 10000. Moreover, filter parameters like permeability, inertial constant and thickness were varied to investigate their effect on the flow upstream of the filter. The following conclusions were drawn from this study: •
First, for Re = 2000 and 3750, FLUENT simulations do not compare well with the experimental results of Yao (2000). Al-Sarkhi (1999) found for slightly different channels that the error in the mean velocity profiles was around 0.5 % and the corresponding accuracy in the Reynolds number was 1% [Yao et al. (2007)]. Hence, these differences may be due to three-dimensionality at these low Reynolds numbers. The results do slightly improve for Re = 3750 (as compared to Re = 2000).
•
However, the re-attachment lengths are predicted well by the Realizable k-ε model (except for Re = 2000).
•
Good agreement between FLUENT and Yao (2000) is observed for Re = 6550 and 10000. FLUENT results also compare well with the experimental observation
88
•
of Armaly et al. (1983) for the case of Re = 7000.Thus FLUENT is able to capture the physics of the re-circulation region to a better extent at higher turbulent Reynolds numbers. The velocity profiles obtained from FLUENT, compared at Re = 6550 and 10000 also re-affirm the trend shown in Yao’s experiments.
•
Finally, when the filter is placed at 4.25 h, at Re = 10000, the variation of permeability (from 1.17*10-7 to 1.17*10-11 m2), inertial constant (4.533*101 to 4.533*106 1/m) and thickness of the filter (from 3.5 mm to 25 mm) have no effect on the re-circulation region upstream of the filter. These may be due to various reasons as discussed in Section 4.2.4.
5.2 Recommendations •
A good low Reynolds number model that uses a damping function to limit the turbulence for better simulations at Re = 2000 and 3750 is recommended for future study.
•
Testing for grid-independence for wall-function approach was found to be cumbersome. Further research using a two-layer approach might be preferable over wall-functions
•
Moreover, the skewness (∆y: ∆x) of some of the grids used in the present study were almost 1:10. This might result in errors that are related to the advection terms in the N-S equations especially in the calculation of X-velocity. However, FLUENT and GAMBIT both considered the present skewness to be within their operating limits. Hence, finer grids along the X-direction can be investigated in future.
89
•
Commercial CFD software FLUENT has not been extensively used in modeling porous media. By modifying the modeling constants from the standard values to values found in the current literature deeper insight can be gained for turbulent flows through porous media and for flows over backward facing step preceding porous media.
•
Further study using different combinations of wall-functions and various models will enable us to decide the best combination (of model and wall-function) for flows with porous inserts.
•
Further elucidation on the boundary Condition that FLUENT uses at the interface of clear-fluid and filter would be appreciated.
•
Spalart-Allmaras model could be tested and compared to k-ε models and it might be a good compromise between desired accuracy and simplicity.
•
The need for more detailed and thorough experimentation on flows with porous inserts would be appreciated in future research.
90
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98
APPENDIX A SNAPSHOTS FROM FLUENT 6.1
The main pull-down menu of FLUENT. Version ‘2ddp’ stands for 2-D Double Precision.
Solver Menu
99
Selection of Turbulence Model in FLUENT; Model constants are kept at default values.
Default properties of air are used.
100
Velocity-inlet Boundary Condition
This particular window shows the case of Re = 10000
101
For porous media, viscous and inertial constants are entered; Flow inside the filter is simulated as laminar [from Yao (2000)]
Residuals are set at 1e-6 (instead of the default value of 1e-3)
102
All PDE’s are at least second-order accurate (AIAA-G077-1998)
Initialization of the simulation is done from inlet
Iteration-start window
103
APPENDIX B
RESULT TABLES
Re = 2000
EXPERIMENTAL RESULTS ( Y/ h ) X/h 1 2 3 3.5 4 5 5.5 5.75 6 6.25 6.5 7 7.24
w/o filter 0.9 0.803 0.798 0.774 0.75 0.64 0.58 0.55 0.52 0.483 0.32 0.22 0.2
filter at 6.75 h 0.883 0.885 0.828 0.8215 0.815 0.767 0.687 0.647 0.607 0.525 0.35
filter at 4.25 h 0.827 0.71 0.426 0.3 0.108
FLUENT RESULTS (Y/ h) X/h 1 2 3 3.5 3.75 4 5
w/o filter 0.65 0.5476 0.4636 0.416 0.38 0.34 0.116
filter at 6.75 h 0.654 0.546 0.46 0.4 0.368 0.328 0.09
104
filter at 4.25 h 0.646 0.524 0.388 0.26 0.112
RESULTS: Re = 3750
EXPERIMENTAL RESULTS ( Y/ h ) X/h 1 2 3 3.5 3.75 4 5 5.75 6 6.25
w/o filter 0.85 0.774 0.705 0.6225 0.54 0.258 0.1
filter at 6.75 h 0.915 0.893 0.876 0.808 0.74 0.59 0.46 0.41 0.376 0.207
filter at 4.25 h 0.84 0.725 0.507 0.2875 0.106
FLUENT (Y/ h) X/h 1 2 3 3.5 3.75 4 5 5.75
w/o filter 0.684 0.572 0.4988 0.4532 0.4272 0.396 0.2452 0.062
filter at 6.75 h 0.6832 0.57 0.492 0.4432 0.4152 0.3832 0.182
105
filter at 4.25 h 0.6736 0.5372 0.404 0.252 0.0812
RESULTS: Re = 6550
EXPERIMENTAL RESULTS ( Y/ h ) X/h 1 2 3 3.5 3.75 4 5 5.75 6
w/o filter 0.89 0.818 0.677 0.59475 0.553625 0.5125 0.404 0.282 0.16
filter at 6.75 h 0.9507 0.883 0.694 0.643 0.6175 0.592 0.435 0.25 0.15
filter at 4.25 h 0.854 0.7 0.42 0.301 0.205
FLUENT (Y/ h) X/h 1 2 3 3.5 3.75 4 5 5.75 6 6.25
w/o filter 0.7272 0.5948 0.5288 0.494 0.4744 0.4528 0.3416 0.2208 0.1608 0.0932
filter at 6.75 h 0.69 0.5772 0.5036 0.4596 0.4348 0.408 0.2296
106
filter at 4.25 h 0.6792 0.5332 0.4004 0.196
Velocity Profiles for Re = 6550 NO FILTER CASE X = 3.75 h 0 -0.29541 -0.3309 -0.32996 -0.32212 -0.30996 -0.29424 -0.27536 -0.25357 -0.229 -0.20177 -0.17196 -0.13962 -0.1048 -0.0675 -0.02773 0.014534 0.059334 0.106728 0.156793 0.209621 0.265318 0.323991 0.385744 0.450667 0.51884 0.590321 0.665144 0.74331 0.824777 0.90946 0.997221 1.08786 1.18111 1.27659 1.37382 1.47216 1.57074 1.66834 1.76323 1.85291 1.93403 2.00279 2.05607 2.09303 2.11564 2.12743 2.13178 2.13063 2.12363 2.10911 2.08557 2.05245 2.01013 1.95942 1.901 1.83515 1.76181 1.6805 1.5904 1.49013 1.37743
0 0.000758 0.001515 0.002273 0.00303 0.003788 0.004545 0.005303 0.006061 0.006818 0.007576 0.008333 0.009091 0.009848 0.010606 0.011364 0.012121 0.012879 0.013636 0.014394 0.015152 0.015909 0.016667 0.017424 0.018182 0.018939 0.019697 0.020455 0.021212 0.02197 0.022727 0.023485 0.024242 0.025 0.025758 0.026515 0.027273 0.02803 0.028788 0.029546 0.030303 0.031061 0.031818 0.032576 0.033333 0.034091 0.034849 0.035606 0.036364 0.037121 0.037879 0.038636 0.039394 0.040152 0.040909 0.041667 0.042424 0.043182 0.043939 0.044697 0.045455 0.046212
X=5h 0 0.500046 0.730264 0.886951 1.01786 1.13411 1.24001 1.33781 1.42872 1.51341 1.59222 1.6653 1.73269 1.79431 1.84986 1.89882 1.94024 1.97284 1.9956 2.00849 2.01277 2.0094 1.99642 1.97008 1.92779 1.8702 1.80092 1.72414 1.64312 1.55999 1.47611 1.39238 1.30943 1.22774 1.14767 1.06948 0.99337 0.919478 0.847906 0.778717 0.711949 0.647617 0.585725 0.526264 0.469212 0.414539 0.362207 0.312175 0.264398 0.218838 0.175464 0.134256 0.095204 0.058308 0.023582 -0.00895 -0.03926 -0.06729 -0.09299 -0.11628 -0.13706 -0.15518
0.05 0.049242 0.048485 0.047727 0.04697 0.046212 0.045455 0.044697 0.043939 0.043182 0.042424 0.041667 0.040909 0.040152 0.039394 0.038636 0.037879 0.037121 0.036364 0.035606 0.034849 0.034091 0.033333 0.032576 0.031818 0.031061 0.030303 0.029546 0.028788 0.02803 0.027273 0.026515 0.025758 0.025 0.024242 0.023485 0.022727 0.02197 0.021212 0.020455 0.019697 0.018939 0.018182 0.017424 0.016667 0.015909 0.015152 0.014394 0.013636 0.012879 0.012121 0.011364 0.010606 0.009848 0.009091 0.008333 0.007576 0.006818 0.006061 0.005303 0.004545 0.003788
FILTER AT 4.25 h X = 3.75 h 0 0.001491 0.085948 0.133585 0.166738 0.19288 0.215128 0.235992 0.257423 0.280821 0.307091 0.336673 0.369611 0.405693 0.444609 0.486077 0.529897 0.575954 0.624198 0.674616 0.727213 0.781996 0.838925 0.897878 0.958665 1.021 1.08437 1.14821 1.21177 1.27386 1.33329 1.38902 1.43965 1.48408 1.52184 1.55269 1.57695 1.59526 1.60831 1.61663 1.6205 1.61996 1.61488 1.60502 1.5902 1.57047 1.54591 1.51661 1.48268 1.44406 1.40081 1.35348 1.30265 1.24855 1.1912 1.13071 1.06733 1.00102 0.931374 0.857861 0.779813 0.696308
107
0 0.000758 0.001515 0.002273 0.00303 0.003788 0.004545 0.005303 0.006061 0.006818 0.007576 0.008333 0.009091 0.009848 0.010606 0.011364 0.012121 0.012879 0.013636 0.014394 0.015152 0.015909 0.016667 0.017424 0.018182 0.018939 0.019697 0.020455 0.021212 0.02197 0.022727 0.023485 0.024242 0.025 0.025758 0.026515 0.027273 0.02803 0.028788 0.029546 0.030303 0.031061 0.031818 0.032576 0.033333 0.034091 0.034849 0.035606 0.036364 0.037121 0.037879 0.038636 0.039394 0.040152 0.040909 0.041667 0.042424 0.043182 0.043939 0.044697 0.045455 0.046212
FILTER AT 6.75 h X=5h 0 0.327293 0.502653 0.625165 0.734335 0.837636 0.936309 1.03088 1.12163 1.20858 1.29167 1.37082 1.4459 1.51676 1.58325 1.64517 1.70222 1.75401 1.79994 1.83925 1.87092 1.89388 1.9076 1.91266 1.90992 1.8985 1.87588 1.83971 1.78962 1.72762 1.65693 1.58064 1.50112 1.42005 1.33859 1.25758 1.17764 1.09925 1.02275 0.94844 0.876517 0.807144 0.740432 0.676458 0.61526 0.556846 0.50119 0.448237 0.397903 0.350088 0.304675 0.261536 0.220536 0.181536 0.1444 0.109006 0.075251 0.043068 0.012449 -0.01653 -0.0437 -0.06882
0.05 0.049242 0.048485 0.047727 0.04697 0.046212 0.045455 0.044697 0.043939 0.043182 0.042424 0.041667 0.040909 0.040152 0.039394 0.038636 0.037879 0.037121 0.036364 0.035606 0.034849 0.034091 0.033333 0.032576 0.031818 0.031061 0.030303 0.029546 0.028788 0.02803 0.027273 0.026515 0.025758 0.025 0.024242 0.023485 0.022727 0.02197 0.021212 0.020455 0.019697 0.018939 0.018182 0.017424 0.016667 0.015909 0.015152 0.014394 0.013636 0.012879 0.012121 0.011364 0.010606 0.009848 0.009091 0.008333 0.007576 0.006818 0.006061 0.005303 0.004545 0.003788
1.2485 1.09682 0.909035 0.631847 0
0.04697 0.047727 0.048485 0.049242 0.05
-0.1704 -0.18235 -0.18988 -0.18201 0
0.00303 0.002273 0.001515 0.000758 0
0.60614 0.507615 0.394196 0.234578 0
0.04697 0.047727 0.048485 0.049242 0.05
-0.09154 -0.11136 -0.12752 -0.13757 0
0.00303 0.002273 0.001515 0.000758 0
Re = 10000
EXPERIMENTAL RESULTS ( Y/ h ) X/h 1 2 3 3.5 3.75 4 5 5.5 5.75 6 6.25 6.5
w/o filter 0.788 0.625 0.535 0.5025 0.48625 0.47 0.405 0.3205 0.27825 0.236 0.216 0.1505
filter at 6.75 h 0.833 0.727 0.6125 0.57775 0.560375 0.543 0.452 0.3475 0.29525 0.243 0.185
filter at 4.25 h 0.747 0.506 0.336 0.222 0.12
FLUENT (Y/ h) X/h 1 2 3 3.5 3.75 4 5 5.5 5.75 6 6.25 6.5
w/o filter 0.6932 0.5856 0.5192 0.4796 0.458 0.436 0.3188 0.256 0.1952 0.1416 0.0768
filter at 6.75 h 0.746 0.6056 0.5376 0.5028 0.4828 0.4608 0.3348 0.2292 0.1468 0.022
108
filter at 4.25 h 0.7396 0.5712 0.442 0.3324 0.2288
Velocity profiles for Re = 10000
NO FILTER CASE X = 3.75 h X=6.25 h 0 -0.41847 -0.48401 -0.49108 -0.4876 -0.47935 -0.46776 -0.45353 -0.43705 -0.41858 -0.3983 -0.37632 -0.35274 -0.32762 -0.30101 -0.27294 -0.24343 -0.2125 -0.18014 -0.14635 -0.11112 -0.07443 -0.03624 0.003466 0.044731 0.087589 0.132095 0.178317 0.226326 0.276202 0.32803 0.381899 0.437903 0.496135 0.556688 0.619645 0.685085 0.753077 0.823679 0.896938 0.972884 1.05153 1.13288 1.2169 1.30355 1.39275 1.4844 1.57836 1.67448 1.77254 1.87228 1.97338 2.07547 2.17809 2.28068 2.38255 2.48286 2.58054 2.67431 2.76261 2.84362 2.91553
0 0.0005 0.001 0.0015 0.002 0.0025 0.003 0.0035 0.004 0.0045 0.005 0.0055 0.006 0.0065 0.007 0.0075 0.008 0.0085 0.009 0.0095 0.01 0.0105 0.011 0.0115 0.012 0.0125 0.013 0.0135 0.014 0.0145 0.015 0.0155 0.016 0.0165 0.017 0.0175 0.018 0.0185 0.019 0.0195 0.02 0.0205 0.021 0.0215 0.022 0.0225 0.023 0.0235 0.024 0.0245 0.025 0.0255 0.026 0.0265 0.027 0.0275 0.028 0.0285 0.029 0.0295 0.03 0.0305
0 0.550998 0.79945 0.961725 1.08913 1.19878 1.29822 1.39119 1.47967 1.56477 1.64715 1.72721 1.80518 1.8812 1.95534 2.02759 2.09792 2.16626 2.23251 2.29656 2.35827 2.4175 2.47405 2.52773 2.57825 2.62532 2.6685 2.70728 2.74094 2.76866 2.78966 2.80364 2.81127 2.81377 2.8119 2.80499 2.79119 2.76832 2.73487 2.69074 2.63736 2.57692 2.51152 2.44286 2.37216 2.30026 2.22778 2.15515 2.08273 2.01076 1.93944 1.86894 1.79939 1.73089 1.66352 1.59735 1.53242 1.46879 1.40648 1.34552 1.28593 1.22771
0.05 0.0495 0.049 0.0485 0.048 0.0475 0.047 0.0465 0.046 0.0455 0.045 0.0445 0.044 0.0435 0.043 0.0425 0.042 0.0415 0.041 0.0405 0.04 0.0395 0.039 0.0385 0.038 0.0375 0.037 0.0365 0.036 0.0355 0.035 0.0345 0.034 0.0335 0.033 0.0325 0.032 0.0315 0.031 0.0305 0.03 0.0295 0.029 0.0285 0.028 0.0275 0.027 0.0265 0.026 0.0255 0.025 0.0245 0.024 0.0235 0.023 0.0225 0.022 0.0215 0.021 0.0205 0.02 0.0195
FILTER AT 4.25 h X = 3.75 h 0 -0.36355 -0.36689 -0.34653 -0.31945 -0.28738 -0.25122 -0.21139 -0.16833 -0.12239 -0.07381 -0.02288 0.030004 0.084461 0.140154 0.196771 0.254008 0.311636 0.369532 0.427662 0.486052 0.544755 0.603845 0.663404 0.723518 0.784272 0.845747 0.908018 0.971156 1.03522 1.10025 1.16628 1.23331 1.30127 1.37009 1.43982 1.51068 1.58262 1.65519 1.72805 1.80099 1.87365 1.94558 2.01626 2.08508 2.15131 2.21411 2.27248 2.32534 2.3726 2.41532 2.45299 2.48446 2.51011 2.53072 2.54698 2.55969 2.56956 2.5772 2.58299 2.58718 2.58986
109
0 0.0005 0.001 0.0015 0.002 0.0025 0.003 0.0035 0.004 0.0045 0.005 0.0055 0.006 0.0065 0.007 0.0075 0.008 0.0085 0.009 0.0095 0.01 0.0105 0.011 0.0115 0.012 0.0125 0.013 0.0135 0.014 0.0145 0.015 0.0155 0.016 0.0165 0.017 0.0175 0.018 0.0185 0.019 0.0195 0.02 0.0205 0.021 0.0215 0.022 0.0225 0.023 0.0235 0.024 0.0245 0.025 0.0255 0.026 0.0265 0.027 0.0275 0.028 0.0285 0.029 0.0295 0.03 0.0305
FILTER AT 6.75 h X=6.25 h 0 0.335705 0.489072 0.587698 0.67335 0.755153 0.834966 0.913487 0.991037 1.06771 1.14347 1.21824 1.29196 1.36449 1.43571 1.50547 1.57361 1.63997 1.70446 1.76701 1.82764 1.88629 1.94286 1.99727 2.04946 2.09941 2.14708 2.19242 2.23537 2.27587 2.31385 2.34915 2.38148 2.41041 2.43554 2.45646 2.47289 2.48471 2.49199 2.49484 2.49329 2.48716 2.47611 2.45956 2.43694 2.40796 2.3727 2.33162 2.28583 2.23664 2.18465 2.13019 2.07397 2.01676 1.95921 1.90173 1.84458 1.78796 1.73205 1.67696 1.62277 1.56953
0.05 0.0495 0.049 0.0485 0.048 0.0475 0.047 0.0465 0.046 0.0455 0.045 0.0445 0.044 0.0435 0.043 0.0425 0.042 0.0415 0.041 0.0405 0.04 0.0395 0.039 0.0385 0.038 0.0375 0.037 0.0365 0.036 0.0355 0.035 0.0345 0.034 0.0335 0.033 0.0325 0.032 0.0315 0.031 0.0305 0.03 0.0295 0.029 0.0285 0.028 0.0275 0.027 0.0265 0.026 0.0255 0.025 0.0245 0.024 0.0235 0.023 0.0225 0.022 0.0215 0.021 0.0205 0.02 0.0195
2.97681 3.02666 3.06529 3.09389 3.11431 3.12842 3.13785 3.14385 3.14723 3.14841 3.14747 3.1441 3.13773 3.12755 3.11273 3.09251 3.06636 3.03405 2.99559 2.95116 2.90104 2.84553 2.78484 2.71907 2.64824 2.57225 2.4909 2.40385 2.31065 2.21065 2.10301 1.98647 1.85921 1.71827 1.55856 1.37071 1.13516 0.785738 0
0.031 0.0315 0.032 0.0325 0.033 0.0335 0.034 0.0345 0.035 0.0355 0.036 0.0365 0.037 0.0375 0.038 0.0385 0.039 0.0395 0.04 0.0405 0.041 0.0415 0.042 0.0425 0.043 0.0435 0.044 0.0445 0.045 0.0455 0.046 0.0465 0.047 0.0475 0.048 0.0485 0.049 0.0495 0.05
1.17088 1.11544 1.06139 1.00872 0.957428 0.907502 0.858929 0.811695 0.765782 0.721173 0.67785 0.635796 0.594996 0.555436 0.517103 0.479986 0.444075 0.409363 0.375845 0.343515 0.312371 0.282413 0.25364 0.226054 0.199658 0.174453 0.150445 0.127636 0.106029 0.085627 0.066434 0.048457 0.03171 0.016228 0.002095 -0.0105 -0.02171 -0.0341 0
0.019 0.0185 0.018 0.0175 0.017 0.0165 0.016 0.0155 0.015 0.0145 0.014 0.0135 0.013 0.0125 0.012 0.0115 0.011 0.0105 0.01 0.0095 0.009 0.0085 0.008 0.0075 0.007 0.0065 0.006 0.0055 0.005 0.0045 0.004 0.0035 0.003 0.0025 0.002 0.0015 0.001 0.0005 0
2.59103 2.59058 2.58835 2.58402 2.57726 2.56768 2.55486 2.53843 2.51815 2.49393 2.46578 2.43378 2.398 2.35856 2.3156 2.26914 2.21899 2.16493 2.10679 2.04439 1.9776 1.90638 1.83076 1.75066 1.66604 1.57716 1.48438 1.38807 1.28878 1.1872 1.08401 0.979852 0.875304 0.770872 0.667141 0.56435 0.455439 0.298232 0
110
0.031 0.0315 0.032 0.0325 0.033 0.0335 0.034 0.0345 0.035 0.0355 0.036 0.0365 0.037 0.0375 0.038 0.0385 0.039 0.0395 0.04 0.0405 0.041 0.0415 0.042 0.0425 0.043 0.0435 0.044 0.0445 0.045 0.0455 0.046 0.0465 0.047 0.0475 0.048 0.0485 0.049 0.0495 0.05
1.5173 1.46609 1.41592 1.36681 1.31875 1.27176 1.22581 1.18088 1.13696 1.09405 1.0521 1.01111 0.971054 0.931903 0.893638 0.85624 0.819692 0.783979 0.749085 0.715 0.681715 0.649224 0.617519 0.586588 0.55642 0.526998 0.498297 0.47029 0.442937 0.416189 0.389986 0.364258 0.338912 0.313831 0.2889 0.264082 0.237588 0.181042 0
0.019 0.0185 0.018 0.0175 0.017 0.0165 0.016 0.0155 0.015 0.0145 0.014 0.0135 0.013 0.0125 0.012 0.0115 0.011 0.0105 0.01 0.0095 0.009 0.0085 0.008 0.0075 0.007 0.0065 0.006 0.0055 0.005 0.0045 0.004 0.0035 0.003 0.0025 0.002 0.0015 0.001 0.0005 0
Separation lines for No-filter case
Experimental X/h 1 2 3 3.5 3.75 4 5 5.5 5.75 6 6.25 6.5 7.24
Re=2000 0.9 0.803 0.798 0.774 0.762 0.75 0.64 0.534 0.481 0.428 0.383 0.32 0.2
Re=3750 0.85 0.774 0.705 0.6225 0.58125 0.54 0.258 0.099 0.1
Re=6550 0.89 0.818 0.677 0.59475 0.553625 0.5125 0.404 0.343 0.282 0.16
Re=10000 0.788 0.625 0.535 0.5025 0.48625 0.47 0.405 0.3205 0.27825 0.236 0.216 0.1505
FLUENT X/h 1 2 3 3.5 3.75 4 5 5.75 6 6.25 6.5 7.24
Re=2000 0.65 0.5476 0.4636 0.416 0.38 0.34 0.116
Re=3750 0.684 0.572 0.4988 0.4532 0.4272 0.396 0.2452 0.062
111
Re=6550 0.7272 0.5948 0.5288 0.494 0.4744 0.4528 0.3416 0.2208 0.1608 0.0932
Re=10000 0.6932 0.5856 0.5192 0.4796 0.458 0.436 0.3188 0.256 0.1952 0.1416 0.0768
Separation lines for filter at 4.25 h
Experimental X/h
Re=2000
Re=3750
Re=6550
Re=10000
1 2 3 3.5
0.827 0.71 0.426 0.3
0.84 0.725 0.507 0.2875
0.854 0.7 0.42 0.301
0.747 0.506 0.336 0.222
4
0.108
0.106
0.205
0.12
FLUENT X/h
Re=2000
Re=3750
Re=6550
Re=10000
1 2 3 3.5
0.646 0.524 0.388 0.26
0.6736 0.5372 0.404 0.252
0.6792 0.5332 0.4004 0.196
0.7396 0.5712 0.442 0.3324
3.75
0.112
0.0812
112
0.2288
Separation lines for filter at 6.75 h
Experimental X/h 1 2 3 3.5 3.75 4 5 5.5 5.75 6 6.25 6.5
Re=2000 0.883 0.885 0.828 0.8215 0.81825 0.815 0.767 0.687 0.647 0.607 0.525 0.35
Re=3750 0.915 0.893 0.876 0.808 0.774 0.74 0.59 0.5 0.46 0.41 0.376 0.207
Re=6550 0.9507 0.883 0.694 0.643 0.6175 0.592 0.435 0.3425 0.25 0.15
Re=10000 0.833 0.727 0.6125 0.57775 0.560375 0.543 0.452 0.3475 0.29525 0.243 0.185
FLUENT X/h 1 2 3 3.5 3.75 4 5 5.5 5.75 6 6.25
Re=2000 0.654 0.546 0.46 0.4 0.368 0.328 0.09
Re=3750 0.6832 0.57 0.492 0.4432 0.4152 0.3832 0.182
113
Re=6550 0.69 0.5772 0.5036 0.4596 0.4348 0.408 0.2296
Re=10000 0.746 0.6056 0.5376 0.5028 0.4828 0.4608 0.3348 0.2292 0.1468 0.022
VITA CHANDRAMOULEE KRISHNAMOORTHY Candidate for the Degree of Master of Science Thesis: NUMERICAL ANALYSIS OF BACKWARD-FACING STEP PRECEEDING A POROUS MEDIUM USING FLUENT
Major Field: Mechanical and Aerospace Engineering Biographical: Personal Data: Born in Pondicherry, India, on May 18, 1982, son of K.Kamakshi and V.Krishnamoorthy Education: Graduated from S.I.W.S. High School and G.N. Khalsa College, Mumbai, India in March 1998 and May 2000 respectively. Received Bachelor of Engineering (B.E.) degree in Mechanical Engineering from University of Mumbai, Mumbai, India in July 2004. Completed the requirements for the Master of Science degree in Mechanical and Aerospace Engineering at Oklahoma State University in August, 2007 Experience: Graduate Trainee Engineer, from July 2004 to May 2005, Blue Star Ltd., India. Teaching Assistant, from Aug 2005 to July 2007, Department of Mechanical and Aerospace Engineering, Oklahoma State University. Professional Memberships: Student Member of American Society of Mechanical Engineers (ASME), Student Member of American Institute of Aeronautics and Astronautics (AIAA).
Name: Chandramoulee Krishnamoorthy
Date of Degree: December, 2007
Institution: Oklahoma State University
Location: Stillwater, Oklahoma
Title of Study: NUMERICAL ANALYSIS OF BACKWARD-FACING STEP FLOW PRECEEDING A POROUS MEDIUM USING FLUENT Pages in Study: 113
Candidate for the Degree of Master of Science
Major Field: Mechanical and Aerospace Engineering Scope and Method of Study: The purpose of the present study is to numerically simulate the flow over a backward-facing step for the no-filter case and for filters at 4.25 and 6.75 step heights. Commercial CFD software FLUENT is used for numerical computations and results are validated with the experimental studies of Yao (2000). The simulations are performed for Reynolds numbers: 2000, 3750, 6550 and 10000. In the present work, GAMBIT software is used for modeling and grid-generation. Different turbulence models in FLUENT namely: SpalartAllmaras, k-ε models, k-ω models and Reynolds Stress model are tested. Amongst the various models, Realizable k-ε model is chosen and gridindependence studies are carried out. The numerical results from FLUENT are then compared with the experimental results of Yao (2000). Moreover, filter parameters like permeability, inertial constant and thickness are varied for the case of Re = 10000 and filter placed at 4.25 step heights. Findings and Conclusions: For Re = 2000 and 3750, FLUENT simulations do not compare well with the experimental results of Yao (2000). The no-filter case at Re = 2000, has a re-attachment length of almost 8 h (step-heights); Turbulent models in FLUENT were unable to simulate this numerically due to transitional nature of the flow at these Reynolds numbers. However, results do slightly improve for Re = 3750. Good agreement between FLUENT and Yao (2000) is observed for Re = 6550 and 10000. Thus FLUENT is able to capture the physics of the re-circulation region to a better extent at higher turbulent Reynolds numbers. The velocity profiles obtained from FLUENT, compared at Re = 6550 and 10000 also re-affirm the trend shown in Yao (2000)’s experiments. Finally, when the filter is placed at 4.25 h, at Re = 10000, the variation of permeability, inertial constant and thickness of the filter have no effect on the re-circulation region upstream of the filter.
ADVISER’S APPROVAL: Frank W. Chambers
