Numerical Simulation of a Blunt Airfoil Wake using a Two-Dimensional URANS Approach Dr Con Doolan∗ School of Mechanical Engineering The University of Adelaide South Australia, 5005 March, 2007
∗
[email protected]
1
Abstract The two-dimensional Unsteady Reynolds Averaged Navier Stokes (URANS) equations have been solved in order to analyse the wake of a blunt airfoil. Turbulence closure was obtained by using the standard twoequation k − ² model and the computed results were compared with experimental data. Only the flow over the trailing edge region was simulated in this study along with the resulting wake flow. A simplified flow domain was constructed using an empirical relation for boundary layer growth on airfoils. A mixed-order grid convergence error estimation method was used and the resulting error of the simulations was estimated to be 2.34%. The simulations show strong vortex shedding and a coherent wake structure that forms after an initial development region. However, the predicted Strouhal number based on the flow velocity measured at a point in the wake was 16.9% below experimental measurements. URANS computations were able to successfully model the effect of vortex shedding on the thickness of the boundary layer at the trailing edge. It was shown that the mean boundary layer profile was significantly changed, with a thickening of the viscous sub-layer increasing the size of the overall boundary layer. Experimental measurements at the edge of the defect-layer compare favourably with numerical results, thus also confirming the suitability of the simplified flow domain for this kind of study. The simulations do not show as good a comparison in the near-wake region of the airfoil and it is believed that this is due to the theoretically ambiguous nature of the URANS approach in areas of sudden flow structure change. Better agreement was observed when measurements were taken after the wake had formed into organised structures however, a persistent difference between numerical and experimental results remained, resulting from poor modelling of the near wake region.
2
Contents 1 Introduction
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2 Flow Domain 2.1 Blunt Airfoil . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Computational Flow Domain . . . . . . . . . . . . . . . . . .
6 6 6
3 Computational Details 3.1 Governing Equations and Solution Method 3.2 Computational Mesh . . . . . . . . . . . . 3.3 Flow Field Initialisation . . . . . . . . . . 3.4 Convergence . . . . . . . . . . . . . . . . .
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4 Simulation Results 15 4.1 Instantaneous Results . . . . . . . . . . . . . . . . . . . . . . . 15 4.2 Streamlines . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 4.3 Mean Flow Results . . . . . . . . . . . . . . . . . . . . . . . . 20 5 Conclusions
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3
1
Introduction
The flow over airfoil trailing edges has occupied the minds of engineers for decades as the concentrated vorticity in the wake induces fluctuating loads and radiated noise which has far-reaching implications for the design of wings, fans and hydrofoils. Further, the interaction of blunt airfoil wakes with other components such as bluff or streamlined bodies represents important flow fields in wind engineering and high-lift devices. Therefore, fundamental understanding and importantly, the ability to predict these types of flows is of central importance to engineers concerned with the design of a wide range of technology. Typically, engineering flows are solved using the steady Reynolds Averaged Navier Stokes (RANS) equations. Unfortunately, using such an approach to investigate the unsteady wake of blunt airfoils is problematic, and probably impossible unless a very coarse mesh and/or highly dissipative numerical method is used that smears the unsteady nature of the flow. An unsteady method is therefore required that resolves the transient nature of the flow field. Direct Numerical Simulation (DNS) is the most accurate method of resolving turbulent unsteady flows but is so computationally expensive that only the simplest flow fields (such as isotropic turbulence in a box or fully developed channel flow) is possible. Large Eddy Simulation is a technique that reduces the computational cost of DNS by using a spatial filter based on the local cell size so that only large turbulent scales (typically in the upper scales of the inertial sub-range) are resolved and the sub-grid-scales are modelled. While the accuracy of LES is well documented in the literature, it is still very computationally expensive and this expense is prohibitive for most engineering flows at moderate to high Reynolds number. Solving the Unsteady Reynolds Averaged Navier Stokes (URANS) equations is therefore attractive form a practical standpoint as it holds the promise of lower computational cost. URANS uses the same turbulence models as RANS but includes the unsteady terms in the governing equations. The lower computational cost occurs as most of the turbulence is modelled and only a small part is resolved. The difficulty with URANS is that it is theoretically ambiguous as it is not really clear how the amount of resolved and modelled turbulence is proportioned within the system of equations. The proportion between modelled and resolved turbulence will vary through major flow changes, such as massive separation, and again there is no clear understanding how URANS adjusts itself to accommodate these changes. Further, the dissipative nature of classical RANS turbulence models (such as the k − ² model) is well known and is due to there development being based on thin shear layers such as boundary layers. Despite these shortcomings, URANS 4
computations are becoming more popular for high Reynolds number engineering flows. This report will investigate the use of two-dimensional (2D) URANS to solve the flow over a blunt, 2D airfoil. It is acknowledged that three-dimensional flow structures play an important role in the development of the wake, but the idea of this study is to investigate the ability of 2D URANS to simulate the wake in an efficient manner. The numerical results are compared with experiments (outlined below) and conclusions drawn at the end of the report.
Background Leclercq [1] has performed a series of experiments in the anechoic wind tunnel at the University of Adelaide. This uses a blunt airfoil to create a stream of turbulent vortices (Von Karman vortex street). In the original experiment, an interacting block was placed in the shed vorticity at various spacings between the vortex generator and the block. An accelerometer was placed within the block to measure flow induced vibration. The block was suspended to allow acceleration to be recorded without significant vibration. Far-field noise was also measured and compared with Curle’s theory [2]. In an effort to improve the original research, a series of experiments were performed that investigated the unsteady wake flow caused by the blunt airfoil. These characterised the flow field about the blunt airfoil trailing edge in more detail. Measurements taken include the Strouhal number, turbulence intensities and wake velocity profiles. These are documented in detail by Souleman [3] and are used for comparison purposes in this study.
Aims As part of a comprehensive simulation effort, the results documented here investigate the vortex shedding process about the trailing edge of the blunt airfoil. The aims of the simulations are: 1. To understand, in greater detail, the flow physics occurring in the wake of the blunt airfoil. 2. To compare the experimental measurements of Strouhal number, boundary layer profiles and mean wake velocity profiles against the computations. 3. Evaluate the effectiveness of the two-dimensional URANS method as a computational tool for this work. 5
Figure 1: Illustration of blunt airfoil used in experiments
2 2.1
Flow Domain Blunt Airfoil
Figure 1 illustrates the blunt airfoil as used in the experiments. The leading edge is elliptical to prevent upstream separation, therefore the only shed vorticity in the flow domain occurs at the trailing edge. The airfoil has a chord of c = 80 mm and a trailing edge thickness of h = 8 mm (c/h = 10). The blunt airfoil has a span of 50 mm which covers the width of the test jet exit of the anechoic wind tunnel. Full details of the flow qualities of the test jet can be found in [4, 3]. Here, it can be stated that the wind tunnel provides a uniform test flow with a turbulence intensity of 0.37%. Additional hot-wire measurements found that that the wind tunnel wall boundary layer height was 1.5 mm for the test conditions used for this study (U∞ = 30 m/s), leaving a large, uniform test core. The Reynolds number based on the trailing edge thickness is Reh = 16000. Velocity measurements were obtained in the wake of the blunt airfoil using a hot-wire anemometer with a single-wire probe. These consisted of point-wise measurements that obtained the Strouhal number of the near-wake as well as wake mean velocity profiles. This information will be compared against the numerical simulations in this report.
2.2
Computational Flow Domain
As upstream separation has been suppressed by using an elliptical leading edge, the main vorticity generation occurs at the trailing edge. Instead of simulating the complete flow about the blunt airfoil, a modified flow domain is used where only the flow over the trailing edge is considered. To obtain 6
Figure 2: Flow domain used for simulations
the correct boundary layer height at the trailing edge, a flat plate boundary layer solution is used. The length of the flat plate is chosen to re-create the boundary layer height expected at the trailing edge of the blunt airfoil. The expected height of the blunt airfoil boundary layer is calculated using the experimental airfoil correlations of Brooks et al. [5]. The correlation for untripped boundary layers is: δ 2 = 10[1.6569−0.9045logRc +0.0596(logRc ) ] (1) c where δ is the boundary layer height, c is the chord and Rc is the Reynolds number based on the chord. Using this correlation, the expected trailing edge boundary layer height can be calculated. Then, using the standard turbulent boundary layer relations [6], an effective length of flat plate can be used to re-create the same boundary layer height. Using this method, it was determined that a flat plate of length 11.2h is required. Therefore, the flow domain consists of a flat prism of length 11.2h and height h, where h = 8 mm. The flow domain is 28.7h × 41h in the x × y directions, as shown in Fig. 2.
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3
Computational Details
3.1
Governing Equations and Solution Method
The unsteady Reynolds Averaged Navier Stokes (URANS) and continuity equations are numerically solved in this study. They are ´ ∂P ∂ ³ ∂Ui ∂Ui =− + + Uj 2νSji − u0j u0i ∂t ∂xj ∂xi ∂xj
(2)
∂Ui =0 ∂xi
(3)
where i, j = 1, 2 (two-dimensional). Further, the specific Reynolds-stress tensor is τij = −u0j u0i
(4)
and the strain-rate tensor is 1 Sij = 2
Ã
∂Ui ∂Uj + ∂xj ∂xi
!
(5)
The velocities (Ui ) represented above are Reynolds averaged, hence the instantaneous velocity is ui = Ui + u0i where u0i is the fluctuating part of the velocity vector. The equations are non-dimensionalised using the trailing edge thickness h and inflow velocity, U∞ . Time is normalised using h/U∞ and pressure using 2 ρU∞ . Trailing edge Reynolds number is therefore Reh = U∞ h/ν where ν is the kinematic viscosity. The Reynolds Averaged Equations are closed using the k −² two-equation turbulence model. The k − ² model provides an estimate of the specific Reynolds-stress tensor via the Boussinesq approximation 2 (6) τij = 2νT Sij − kδij 3 where νT is the kinematic eddy viscosity and k is the turbulent kinetic energy. The eddy viscosity is estimated using νT = Cµ k 2 /²
(7)
Separate equations are solved for the turbulent kinetic energy and dissipation rate (²) respectively
8
"
∂k ∂k ∂ ∂k ∂Ui + Uj = τij −²+ (ν + νT /σk ) ∂t ∂xj ∂xj ∂xj ∂xj
#
"
² ∂Ui ²2 ∂² ∂ ∂² ∂² = C²1 τij − C²2 + + Uj (ν + νT /σ² ) ∂t ∂xj k ∂xj k ∂xj ∂xj
(8) #
(9)
The following closure coefficients are required to complete the set of equations C²1 = 1.44 C²2 = 1.92 Cµ = 0.09 σk = 1.0 σ² = 1.3
(10)
A standard no-slip wall boundary condition is applied to the surface of the trailing edge. As the Reynolds number is reasonably high, wall functions are used to set the velocity boundary conditions close to the surface of the trailing edge. These use the well known relation for the log-layer region of the turbulent boundary layer 1 ³ +´ ln Ey κ U = u sτ τw = ρ yuτ = ν
u+ =
(11)
u+
(12)
uτ y+
(13) (14)
where κ = 0.4187 and E = 9. Using these relations, the values for turbulent kinetic energy and dissipation can be set at the cell closest to the surface (at y = yp ) u2 kp = q τ Cµ ²p = Cµ0.75
(15) k 1.5 κyp
(16)
It has been shown (e.g. [7]) that wall functions are a suitable method to reduce the meshing requirements near wall while maintaining accuracy provided the mesh spacing near surface satisfies 10 < ∆yn+ < 100 9
(17)
Table 1: Mesh density and ∆yn+ values Mesh Very Coarse Coarse Medium Fine Very Fine
Nx × Ny 58 × 34 115 × 65 163 × 91 230 × 130 325 × 184
Maximum ∆yn+ 149.8 102.4 80.9.4 56.4 33.8
Average ∆yn+ 20.4 17.5 15.2 12.2 9.5
Note that ∆yn+ represent the mesh spacing normal to the surface in the x or y directions. Additionally, if the mesh spacing is sufficiently refined so that ∆yn+ < 10, then the wall function for the viscous sub-layer is used instead of the log-layer u+ = y +
(18)
This set of equations are discretised using a structured finite-volume method [8]. The convective and diffusive terms are evaluated using a secondorder accurate central-differencing method. Time integration is performed using an Euler method with the requirement that the maximum Courant number is kept below 0.2. The pressure-implicit split-operator (PISO) algorithm with two correction steps is used as an implicit, transient solution scheme. The resulting system of equations are solved using using the incomplete Choleski conjugate gradient method with a solution tolerance of 10−6 .
3.2
Computational Mesh
The simulation described here was performed on a non-uniform Cartesian mesh. Five meshes were used to establish the accuracy of the solution method, although the Very Coarse and Coarse Meshes are only listed for completeness and was not used as part of the error estimation process as listed in Section 3.4. These are listed in Table 1 which lists the mesh density, maximum and average ∆x+ or ∆y + values adjacent to the walls of the trailing edge. These values were taken after the initial flow field transients had disappeared.
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3.3
Flow Field Initialisation
A two-dimensional potential flow solver is used to create a conservative flow field to initialise the simulations. Using this as the starting condition at tU∞ /h = 0, the two-dimensional URANS equations are iteratively solved. It takes approximately 5000 (very fine mesh) time steps or a non-dimensional time of 18.75 for vortex shedding to begin. This is illustrated in Fig. 3, which shows the normalised velocity magnitude versus non-dimensionalised time at a position x = 1.75h, y = 0.75h from the trailing edge mid-plane for all computational meshes used in this study. This corresponds to a point where experimental data was taken by hot-wire anemometer. Figure 3 shows the solution progressing from the initial field as determined by the potential flow solutions to the point where data sampling begins (tU∞ /h = 150). It can be seen that initial flow transients have disappeared by tU∞ /h = 150 or 5.2 computational domain flow-through times. Flow simulations are obtained after the initialisation period for tU∞ /h = 112.5 non-dimensional time units or 4 domain flow-through times. This corresponds to 30000 time steps and captures 25 vortex shedding cycles with 1200 time steps per shedding period on the finest mesh.
3.4
Convergence
Determining the grid convergence or discretisation error is an important part of any numerical analysis. Unfortunately, there are a variety of techniques available in the literature for this purpose and it is sometimes confusing to choose a reliable method to estimate convergence error. Most methods are based on a Richardson extrapolation [9] where order of the solution is assumed in advance based on the discretisation schemes used to solve the equations. However, in the case of turbulent flows using turbulent closure models (such as the k − ² model used here), studies [10, 11] have shown that the method to determine the order of the solution can give spurious results, especially in shear layers and across shock waves. Therefore, it has been recommended that the order of solution be bounded by realistic values as dictated by the actual discretisation schemes used. In the current flow simulations, all spatial derivatives are approximated by second order accurate schemes, but the time integration is Euler, which is first order. Therefore, the method is regarded as mixed and the order should be realistically bounded between 1 and 2. Celik and Karatekin [11] apply bounded first and second order Richardson extrapolation to a turbulent flow simulation case for flow over a backward 11
1.4
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(e) Very Coarse Mesh
Figure 3: Resolved velocity magnitude at x = 1.75h, y = 0.75h during simulation startup
12
facing step. An estimate of the exact, grid independent solution is found using the following extrapolation formula rp φh − φrh (19) rp − 1 The grid-refinement-ratio, r, is the ratio of mesh sizes that are compared. In each mesh considered here, r = 2 as each mesh density is doubled from medium-to-fine-to-very fine. The term φ represents a particular solution variable used to estimate convergence. For first-order solutions, p = 1 and for second-order solutions, p = 2. The subscript h refers to the finest mesh, rh to the medium mesh and r2 h to the coarsest mesh. Roy [10] developed a mixed-order grid convergence error estimation method. In this method fexact =
fexact = φh +
²32 − 5²21 3
(20)
where ²ij = φi − φj
(21)
where i, j = 1, 2, 3 corresponding to the finest (1), medium (2) and coarsest (3) mesh. To estimate the error in the numerical solution, the finest mesh solution is compared with the exact solution obtained from the first, second or mixedorder methods above. This is done using eh =
fexact − fh fh
(22)
A conservative estimate of discretisation error is the Grid Convergence Index, or GCI [12]. This introduces a safety factor when calculating the discretisation error GCI = Fs
fexact − fh fh
(23)
where Fs = 3 is the usual quoted value but Wilcox [7] recommends using Fs = 1.25 for three-mesh convergence studies. For the current study, the non-corrected error (Fs =1) will be used (following Roy [10]) to estimate discretisation error using the first, second and mixed-order methods. To estimate discretisation error, the time-varying flow velocity at a point x = 1.75h, y = 0.75h from the trailing edge mid-plane is used for the analysis. Two flow parameters are calculated, the Strouhal number (St = f h/U∞ ) and 13
Table 2: Mesh Convergence Results Mesh Coarse Medium Fine Very Fine fexact (Mixed-Order) [10] fexact (First-Order) [11] fexact (Second-Order) [11]
Ntot 7475 14833 29900 59800 -
St = f h/U∞ 0.231 0.227 0.222 0.222 -
Urms /U∞ 1.122 1.171 1.133 1.110 1.084 1.087 1.102
Table 3: Discretisation Error Results Order Error, eh Mixed-Order 2.34% First-Order 2.07% Second-Order 0.73%
the normalised RMS flow velocity (Urms /U∞ ). Table 2 summarises the results from the convergence study. Also shown are the grid independent or ‘exact’ values calculated using schemes assuming different order. It is interesting to observe that the mixed order scheme of Roy [10] is outside the first and second order extrapolation estimates. It can be seen that the Strouhal number converges monotonically and reaches an error level that is near the machine error on the very fine grid. It is therefore believed that this value is accurate and will not be used for the convergence study. The normalised velocity is more interesting to observe. It can be seen that the velocity does not converge monotonically. This type of behaviour is associated with mixed convergence schemes and is noticed in other RANS-type flow solutions [11]. Table 3 summarises the error calculated using the first, second and mixed order schemes. It can be seen that the mixed-order method gives the highest error and, for the reasons outlined above, it will be used as the error estimate for this study.
14
Figure 4: Contours of instantaneous vorticity magnitude at tU∞ /h = 262.5
4 4.1
Simulation Results Instantaneous Results
Figure 4 shows colour contours of instantaneous vorticity about the trailing edge region at tU∞ /h = 262.5. In this and subsequent figures, the length scale is given by the trailing edge height h = 8 mm. It can be seen that a VonKarman-like vortex street is formed behind the trailing edge, due to strong, alternate shedding of the airfoil boundary layers from the top and bottom surfaces. The shedding process is complex, as it involves a redistribution of eddy length-scales from the boundary layer, where the maximum energy containing eddies have a size lp ∼ δ, to the wake, where lp ∼ h. The unsteady nature of the wake induces fluctuating forces on the airfoil which results in radiated noise (as measured in experiments [1, 3]). Figure 5 shows colour contours of the instantaneous flow velocity magni15
Figure 5: Contours of instantaneous velocity magnitude at tU∞ /h = 262.5
tude in the trailing edge region at tU∞ /h = 262.5. The results show an initial wake development region, which extends over a length of approximately 3 − 4h in the x-direction. After this initial region, the wake forms periodic flow-structures (large eddies) that convect downstream and out of the flow domain. Similarly, Fig. 6 shows colour contours of instantaneous pressure at at tU∞ /h = 262.5. The pressure results also clearly show the formation of an unsteady wake consisting of coherent flow structures convecting in the direction of the mean flow. It may be seen that the pressure field influences the flow a considerable distance upstream of the trailing edge. It is expected that the fluctuating pressure and velocity at the trailing edge will affect the mean turbulent boundary layer structure, and this will be investigated later in this report. The time history of the velocity taken at a point just behind the trailing edge at x = 1.75h, y = 0.75h from the trailing edge mid-plane is shown 16
Figure 6: Contours of instantaneous pressure at tU∞ /h = 262.5
17
Table 4: Strouhal number comparison URANS 0.222 Experiment 0.267
in Fig. 7(a). It can be seen that the signal is periodic, with a normalised frequency, or Strouhal number, of f h/U∞ = 0.222. This is confirmed by the frequency spectra shown in Fig. 7(b), which was calculated using a FastFourier-Transform (FFT) procedure with no pre-treatment or filtering of the numerical data. Table 4 compares the Strouhal number obtained by experiment [3] with the fundamental Strouhal number for the URANS computation. The simulation under predicts the Strouhal number by 16.9%. Figures 8 and 9 show the modelled turbulence kinetic energy and dissipation, respectively. It is important to note that, in an unsteady simulation, the turbulent kinetic energy consists of resolved and modelled components, in an analogous manner to the filtered and sub-grid-scale terms obtained during a Large-Eddy-Simulation. For example, following Pope [13], the kinetic energy of the fluid per unit mass is 1 E(x, t) ≡ U(x, t) · U(x, t) (24) 2 ¯ therefore consists of three parts, the kinetic The mean kinetic energy (E) ¯ the resolved turbulent kinetic energy (kr ) and energy of the mean flow (E), the modelled turbulent kinetic energy (km ) ¯ ¯ t) + kr (x, t) + km (x, t) E(x, t) = E(x,
(25)
where 1¯ ¯ U·U E¯ ≡ 2 1 0 0 k ≡ uu 2 i i
(26) (27)
The modelled kinetic energy is obtained using the turbulent kinetic equation (Eq. 8) while the resolved kinetic energy is obtained using using the resolved velocity fluctuation (u0r ), directly from the computed URANS flow field. That is, u0r = ur − U¯ , where ur is the resolved velocity and U¯ is the mean velocity. Thus, it is important that the amount of kinetic energy that is modelled and resolved be correct for the particular flow situation. Unfortunately, 18
1.4 1.3 1.2
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60
10log10[F(|U|)] (dB)
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(b) Frequency spectra
Figure 7: Velocity data obtained at a point x = 1.75h, y = 0.75h from the trailing edge mid-plane
19
the theoretical basis for how this split occurs in URANS computations is ambiguous and leads to some erroneous results, particularly in regions where the flow field is undergoing severe structural change, such over a blunt trailing edge. Returning to Figs. 8 and 9, it can be seen that immediately after the trailing edge, there is major structural change in the flow field, as the boundary layer shear flow changes into a wake flow. As shown in previous results, the modelled turbulent kinetic energy and dissipation undergo reorganisation over a distance of 3 − 4h after the trailing edge. After this region, the flow starts to achieve a more ordered state that is more accustomed to a two-dimensional wake flow. It is important to accurately model the nearwake region if downstream wake features (such as spreading rate) are to be accurate also, as turbulent scales are long-lived and have considerable downstream influence. As the turbulence models used here are based on steady shear flow (boundary layers), it is expected that the turbulent viscosity will be incorrect in the near-wake. It is believed that this is one of the causes for the lower computed Strouhal number when compared with experiment. The lack of modelling spanwise eddy structures will also influence this as well. More on this later in the report when wake simulations are compared with experimental data.
4.2
Streamlines
Figure 10 shows mean (a) and instantaneous streamlines computed from the numerical results. The mean streamlines show a well-defined, symmetric recirculation region immediately behind the trailing edge. The instantaneous streamlines show the distortion of the recirculation region in the near wake. Also, the streamlines reveal the oscillating nature of the wake flow after an initial formation zone, consistent with previously discussed results.
4.3
Mean Flow Results
Boundary Layer Numerical and experimental mean boundary layer velocity profiles are compared at the location of the trailing edge. For these results, y = y + = 0 indicates the surface of the trailing edge. It can be seen that the vortex shedding has a strong effect on the mean boundary layer size. Compared against the results are profiles expected for the viscous sub-layer (u+ = y + ) and the log-layer (u+ = 1/κ ln y + + 5.35). Vortex shedding has the effect of thickening the viscous sub-layer (y + < 25) and slightly modifying the slope 20
Figure 8: Contours of instantaneous modelled turbulence kinetic energy at tU∞ /h = 262.5
21
Figure 9: Contours of instantaneous modelled turbulence dissipation at tU∞ /h = 262.5
22
(a) Mean flow field
(b) Instantaneous flow field at tU∞ /h = 262.5
Figure 10: Streamlines in trailing edge region
23
35
URANS computation Law of the wall: viscous sub layer Law of the wall: log-layer Experiment
30 25 +
+
u =y
u
+
20
u+ = (1/κ)ln y+ + 5.35
15 10 5 0 1
10
100
1000
10000
+
y
Figure 11: Comparison between time averaged URANS computation, experiment and law of the wall
of the log-layer (25 < y + < 200). URANS computations also show that the mean velocity level falls as y + is increased into the defect-layer (y + > 200), where the usual expectation is that the u+ values will rise. Hence vortex shedding appears to have the effect of greatly increasing the thickness of the lower regions of the boundary layer and distorting the merging of the boundary layer with the free stream. Experimental validation of these results are possible using available data. Single component hot-wire data [3] was obtained to the edge of the defectlayer and confirms that the mean boundary layer thickness as calculated by URANS is accurate. Therefore, it can be concluded that the URANS method can accurately resolve mean velocity information for thin shear layers. Also, the technique used to simplify the flow domain is shown to be accurate as well. Wake Flow Mean velocity experimental data [3] for the wake region is also compared against URANS numerical results in Fig. 12. Experimental and numerical 24
results are compared at three downstream locations: x/h = 1.75, x/h = 4.88 and x/h = 8.00. In the near-wake region (x/h = 1.75), agreement is poor, with the URANS results overestimating the velocity defect behind the trailing edge. As discussed earlier, it is believed that the modelling of turbulent production and dissipation is not correct in the near wake region of a URANS computation. The very good agreement in the boundary layer confirms the basis by which the k − ² turbulence model has been developed - on shear flows where turbulent viscosity can be reasonably expected to be constant. This is true for boundary layers and also for self-similar shear flows of other descriptions [13]. In the near-wake, major flow changes are occurring and the assumption that production and dissipation are matched (as occurs in boundary layer flows) breaks down, resulting in poor comparisons with experiment. Physically, the URANS solution method over predicts the turbulent eddy viscosity in the near wake region. At distances beyond the initial development region indicated in the colour contour plots (x/h = 4.88 and x/h = 8.00), agreement is better, but there is a persistent error, most likely due to poor resolution of the flow field in the near wake. It is easy the comprehend that, in a temporally and spatially evolving field, that any error in the initial conditions will result in that error propagating downstream an persisting for large distances.
5
Conclusions
A 2D URANS simulation of the flow over a blunt airfoil has been performed and compared with experimental data. In summary, the numerical method captures the important flow physics of the unsteady wake and the following major conclusions can be drawn: 1. Simplifying the flow domain using an empirical airfoil boundary layer relationship has been shown to be successful for this type of flow. 2. 2D URANS can accurately model the effect of vortex shedding on the mean turbulent boundary layer profile at the trailing edge of a blunt airfoil. 3. 2D URANS will simulate periodic vortex shedding but at a lower frequency than the experimental measurement. 4. The mean velocity defect in the near wake is significantly over-predicted using 2D URANS.
25
4 3
URANS computation Experiment x/h = 1.75
x/h = 4.88
x/h = 8.00
2
y/h
1 0 -1 -2 -3
|U|/U∞= 1 |U|/U∞
Figure 12: Comparison between time averaged URANS computation and experiment at various wake cross-sections
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5. The agreement between computation and experimental mean velocity wake profiles are in better agreement when coherent structures are well formed in the wake, however a persistent difference is maintained due to the poor resolution of the near wake. As a final note, it is concluded that 2D URANS is a useful tool for studying the flow physics as long as the user keeps their eyes open and understands the limitations of using this solution method. To increase the accuracy of URANS methods, more attention is required to increase the fidelity of the turbulence closure model and understanding how to control or better influence the distribution of turbulent kinetic energy between modelled and resolved components. One such approach is to develop a solution technique for the Temporally Filtered Navier Stokes (TeFiNS) equations that incorporates an understanding of the resolution ability of the mesh and frequency limits of the k − ² model.
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[8] Jasak, H., Weller, H., and Nordin, N., “In-Cylinder CFD Simulation using a C++ object orientated toolkit,” Tech. Rep. SAE Technical Paper 2004-01-0110, Society of Automotive Engineers, 2004. [9] Richardson, L., “The Approximate Arithmetic Solution by Finite Differences of Physical Problems Involving Differential Equations with Application to Stresses in a Masonry Dam,” Transactions of the Royal Society of London, Series A, Vol. 210, 1910, pp. 307–357. [10] Roy, C., “Grid Convergence Error Analysis for Mixed-Order Numerical Schemes,” AIAA Journal , Vol. 41, No. 4, 2003, pp. 595–604. [11] Celik, I. and Karatekin, O., “Numerical Experiments on Application of Richardson Extrapolation with Nonuniform Grids,” Journal of Fluids Engineering, Vol. 119, September 1997, pp. 584–590. [12] Roache, P., Verification and Validation in Computational Science and Engineering, Hermosa Publishers, Albuquerque, NM, USA, 1998. [13] Pope, S., Turbulent Flows, Cambridge University Press, Cambridge, UK, 2000.
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