REYNOLDS-AVERAGED NAVIER-STOKES ... - Semantic Scholar

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REYNOLDS-AVERAGED NAVIER-STOKES CALCULATIONS OF UNSTEADY TURBULENT FLOW H. L. Zhang, C. R. Bachmany and H. F. Faselz Department of Aerospace and Mechanical Engineering, University of Arizona Tucson, Arizona, USA

Abstract

all signi cant scales of the uid motion are resolved without using any turbulence models [1]. For the foreseeable future, DNS will remain limited to relatively low Reynolds number ows and/or to simple geometries because of the high demands for computer resources to resolve the smallest scales (down to the Kolmogorov scale). Therefore, DNS is not practical for many engineering applications. In contrast, in Large Eddy Simulations (LES), simple turbulence models are used to model the smallest scales, while the three-dimensional, time-dependent motions of the large scales are computed. Hence, the smallest grid cell required for resolving the smallest vortical structures can be much larger than the Kolmogorov scale, which, therefore, greatly reduces the requirement for computer resources. Both DNS and LES are useful to investigate the nature of turbulence ows and can aid in the development of turbulence models used for Reynolds Averaged Navier-Stokes (RANS) calculations. LES, in some cases, can be employed to predict ows of technical interest. However, the great bulk of engineering calculations of turbulent ows are still beyond the realm of both DNS and LES. Instead, RANS calculations are predominantly used in industry for prediction of turbulent ows. Traditionally, in RANS calculations of turbulent

ows, only a steady-state solution is of interest, since the time-averaged Navier-Stokes equations are solved. However, as the time interval used for the time averaging of the Navier-Stokes equations is decreased, and the transport equations for the turbulent quantities (such as the kinetic energy and dissipation rate) are solved in a time-dependent fashion, a so-called Unsteady RANS (URANS) calculation can be obtained. It is hoped that in URANS calculations the intrinsic, large unsteady vortical structures that are physically present in the ow eld under consideration will be captured by the calculations, in spite of the dissipative nature of the turbulence models used in RANS calculations. An example in the literature of URANS applications is the calculation of the dynamic stall of a pitching aerofoil using various turbulence mod-

In this study, a combination of the unsteady incompressible Navier-Stokes equations in vorticityvelocity formulation and the Algebraic Stress Model (ASM) of Gatski and Speziale (1996) is employed for Unsteady Reynolds Averaged Navier-Stokes (URANS) calculations of turbulent boundary layer

ows. The Navier-Stokes equations are solved using a fourth-order compact dierence scheme in space and a fourth-order Runge-Kutta method in time. The highly accurate numerical method greatly reduces the possibility of contamination of the results by second-order arti cial dissipation from the numerical schemes. A at plate boundary layer subjected to a strong adverse pressure gradient with laminar separation and turbulent reattachment is investigated. Performing URANS calculations for this ow, we found that unsteady vortical structures remain in the ow eld despite the large \eective eddy viscosity" produced by the turbulence model (ASM). This is due to the fact that a special function is used in this turbulence model such that the eddy viscosity is strongly coupled with the unsteady

ow structures. For comparison, URANS calculations were also carried out employing the standard k ?  model, where in contrast no unsteady vortical structures were found in the ow eld. For further comparison, results from 2-D \Direct Numerical Simulation (DNS)" and 3-D Large-Eddy Simulation (LES) using the standard Smagorinsky model are also presented and discussed.

1 Introduction In Direct Numerical Simulations (DNS), threedimensional, time-dependent numerical solutions of the Navier-Stokes equations are obtained such that  Research Associate. Member y Research Assistant. z Professor, Aerospace and

AIAA.

Mechanical Engineering. Member AIAA. c 2000 by American Institute of Aeronautics Copyright and Astronautics, Inc. All rights reserved.

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els, such as the k ? ! model (Liu and Ji [2]), the Baldwin-Barth model (Guilmineau et al. [3]), the one-equation model (Ekaterinaris et al. [4]) and ve other models (Srinivasan et al. [5]). Other examples include the unsteady ow around a circular cylinder [6], behind a cylinder with triangular cross section [7], the ow over a backward-facing step [8] and the ow in a diuser with a rapid expansion [9]. For these examples, the k ?  model and the JohnsonKing model, respectively, were employed. To a certain degree, the numerical results agree qualitatively with the experimental measurements, including the experimentally observed large vortical structures. However, in most of the cases mentioned above, numerical schemes with low (up to second) order accuracy were applied for solving the NavierStokes equations. Low order schemes can contaminate the numerical results due to arti cial diusion. In addition, either an external unsteadiness (like oscillating aerofoils) or a massive separation in the

ow eld (like a backward-facing step) was taken as validation cases. These cases were somewhat less critical to test the capability of turbulence models with regard to their ability to capture the intrinsic unsteadiness due to the presence of the vortical structures in the ow eld. The motivation of the present study is (i) to use numerical schemes with high (at least fourth) order accuracy to solve the Navier-Stokes equation in order to reduce the contamination of the applied turbulence model by second-order arti cial dissipation; and (ii) to test the behavior of different turbulence models. As a test geometry, a

at plate boundary layer subjected to a strong adverse pressure gradient was chosen, where laminar separation and turbulent reattachment occur and unsteady vortices are shed into the turbulent boundary layer downstream of the reattachment location. The turbulence models used in this study include the Algebraic Stress Model (ASM) by Gatski and Speziale [10] and the standard k ?  model. For comparison, this ow was also calculated using a two-dimensional (2D) \DNS" and a three-dimensional (3D) LES based on the standard Smagorinsky model. In this paper, the numerical method and the various turbulence models used in the study are presented. Validation calculations for a turbulent at plate boundary layer without pressure gradient and with an increasingly adverse pressure gradient are discussed. Then results for a boundary layer subjected to a strong adverse pressure gradient with a laminar separation and turbulent reattachement

are presented.

2 Numerical Method 2.1 Governing equations The governing equations are the incompressible, unsteady Navier-Stokes equations in vorticity-velocity formulation [11]. After Reynolds averaging the Navier-Stokes equations and subsequently applying the curl operator, three transport equations for the averaged vorticity are obtained

@~! + r  ~!  V~ ? r
~  = 1 r2 ~! ; @t Re

(1)

where the overbar on the velocity vector V~ and the vorticity vector ~! = ?r  V~ denotes the Reynolds averaging of the appropriate components of the corresponding vector. ~ = [ij ] is the Reynolds stress tensor. Equation (1) is normalized by the freestream velocity U1 , a reference length L. A global Reynolds number is de ned as Re = U1L= , where  is the kinematic viscosity. In addition to the transport equations for vorticity, the governing equations also include a set of Poisson type equations for the velocities. From the de nition of vorticity and using the fact that both the velocity and the vorticity elds are solenoidal, three equations for the velocity components are obtained (see [11] for more details). Thus, a complete system of governing equations is obtained, except that the Reynolds stress tensor must be modeled (see section 2.2 for more details). Using the unsteady Reynolds-averaged NavierStokes equations for calculating unsteady turbulent

ows in this study will be referred to in the remainder of this paper as URANS calculations (for Unsteady RANS).

2.2 Turbulence models In the present study, the Algebraic Stress Model (ASM) proposed by Gatski and Speziale [10] was implemented to demonstrate the capability of URANS calculations for capturing large unsteady vortical structures in the ow eld. For comparison, URANS calculations using the standard k ?  model were also performed. 2

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2.2.1 Algebraic Stress Model

2.2.2 Standard k ?  model

In the Algebraic Stress Model (ASM) [10] the Reynolds stress tensor are modeled as 2 ij = ijASM = 23 kij ? 1 f (;  ) k S ij + ijA ; (2) where the strain tensor are given by   @ui + @uj ; S ij = 21 @x (3) j @xi and f (;  ) = 3 ? 232 + 6 2 2

with

+ )  3 + 23(1 + 62  2 + 6 2 ;

 = 21 3 (S ij S ij )1=2 k ; 1  2  =  (!i !i )1=2 k : 1

In order to validate the ASM results, calculations were also performed using the standard k ?  model [12, 13]. Traditionally, with the Boussinesq assumption, the Reynolds stress tensor in equation (1) can be written as (12) ij = ijk- = 32 kij ? 2T S ij ; where T is the eddy viscosity as before (equation (9)) and k and  are also obtained from equation (7) and (8) as before, except that the wall damping functions are now f = 1 ? exp(?y+=A+ ); (13)

(4)

p

f2 = 1 ? exp(?y Re k=10);

(5)

(14)

where A+ = 25. Comparing equation (2) with (12), it can be seen that 1 f (;  )k= is equivalent to 2T .

(6)

Therefore, in this study of the ASM model, the former is called the \eective eddy viscosity".

In equation (2), ijA represents the anisotropic part

of the subgrid scale stress tensor (neglected in the present study). The quantities k and  are the turbulent kinetic energy and dissipation rate, respectively, which are governed by the following transport equations:

2.2.3 Smagorinsky model (LES)

In contrast to RANS calculations where time averaging is applied to the governing equations, in LES spatial ltering is applied to the Navier-Stokes equations to derive the resolved-scale equations. The same form of vorticity transport equations can be obtained for Large Eddy Simulations (LES) as equation (1). However, [ij ] now represents the Leonard, cross-term and sub-grid scale (sgs) Reynolds stresses. The fundamental problem of LES, just as for RANS methods, is to establish a model for these stresses so that the governing equations can be closed. Smagorinsky [14] was the rst to postulate a simple model for the sgs stresses. The model assumes that the sgs stresses follow a gradient-diusion process, similar to molecular motion. Thus, ij = ijLES = ?2T S ij ; (15) where T is the eective viscosity of the subgrid scales, q T = l2 2S ij S ij ; (16) where  +=A+ )3  ? ( y (
x
y
z )1=3 (17) l = Cs 1 ? e

@k + (V~
r)k = ? @ui ?  ij @x @t   j @k + 1 r2 k; (7) + @x@ T @x Re j k j @ + (V~
r) = ?C   @ui ? C f 2 2 2 k 1 k ij @x @t j   @ + 1 r2 ; + @x@ T @x (8) Re j  j where T is the eddy viscosity, T = C f k2 = (9) and f and f2 are the wall damping functions, f = (1 + p3:245 )[1 ? exp(?y+ =70)]; (10) k = f2 = [1 ? exp(?y+ =4:9)]2: (11) The constants needed for solving the k ?  equations (7) through (10) are k = 1;  = 1:3; C = 0:09; C1 = 1:44; C2 = 1:83; while the coecients in equations (2), (5) and (6) are: 1 = 0:227; 2 = 0:0423; 3 = 0:0396:

and Cs = 0:065 is the Smagorinsky constant.
x ,
y and
z are the grid widths in the x, y and z directions, respectively. 3

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2.3 Numerical scheme

case of an adverse pressure gradient, @v=@y is speci ed, which is equivalent to prescribing @u=@x or @p=@x via the continuity equation and the Bernoulli equation, respectively. At the out ow boundary, x = xmax , the second derivatives with respect to x of all variables are set to zero. For 3D calculations, periodicity is assumed in the spanwise z -direction. With these boundary conditions, the governing equations were solved inside an integration domain x0  x  xmax , 0  y  ymax (and 0  z  zmax for 3D simulations), using the dierence schemes described above (see section 2.3 for more detail).

A detailed description of the numerical scheme used in this study for solving the governing equations is given by Meitz and Fasel [11]. The main features of the numerical scheme include the following: (i) The streamwise and wall-normal derivatives are approximated by compact dierences of fourth order accuracy; (ii) for 3D simulations, the ow is assumed to be periodic in the spanwise direction z , therefore, this direction is treated pseudo-spectrally; (iii) a fourth-order Runge-Kutta method is used for the time integration; (iv) a fast Poisson solver is implemented for solving the velocity equations; and (v) at the out ow boundary a buer domain technique is implemented to avoid re ections of disturbance waves and vortical structures [15]. For the ASM as well as the standard k ?  model, equations (7) and (8) are solved using a second-order accurate numerical scheme, and an ADI method was utilized to advance in time.

3 Steady RANS Calculations The code and the implemented turbulence models (the ASM and the standard k ?  model) have been validated rst for a 2D turbulent boundary layer on a at plate with zero pressure gradient. In the x ? y plane, a grid with 341128 points was used for the 2D calculations. The rst grid point next to the wall was at about y+ = 2. The freestream velocity was 15 m=s, while the reference Reynolds number was 106=m. At the in ow boundary as mentioned before, an empirical solution of the 1/7th law was imposed for the streamwise velocity, while the normal velocity was obtained from integrating the continuity equation. Since this solution does not exactly satisfy the Navier-Stokes equations, there is a strong adjustment of the Navier-Stokes solutions near the in ow in the computational domain, as can be observed in the skin friction distribution in Figure 2(a). Eventually, the calculated values of the skin friction for both the ASM and the standard k ?  model will approach the theory [17], although for the standard k ?  model a little undershooting occurs in the downstream direction. For the displacement thickness and the momentum thickness, the results from both models agree well with the theoretical predictions (see gure 2(b)). Turbulent kinetic energy k and dissipation rate  are dicult to predict, since they are strongly dependent on the upstream history and the boundary conditions. In the present study, a distribution of k and  from measurements was interpolated at the in ow boundary as xed in ow conditions, while two types of wall boundary conditions for  have been tested:

2.4 Boundary conditions

A schematic of the computational domain as well as of the experimental setup by Gaster [16] are shown in gure 1. For the in ow boundary at x = x0 , a Airfoil Flow

Inflow

Free Stream

Buffer Domain

Out− flow

Flat Plate

Laminar

Separation

Turbulent

Figure 1: A schematic of the ow domain over a

at plate and the experimental setup from Gaster (1966). Blasius solution is speci ed if the ow is assumed to be laminar. On the other hand, if the incoming ow is turbulent, an empirical 1/7th law for the streamwise velocity pro le is used, while the k and  distributions are interpolated from measurements. In addition, all x derivatives needed for the compact dierence approximations of the governing equations are also speci ed. At the wall (y = 0), noslip, non-permeable wall conditions are imposed for the velocity components. The boundary conditions for k and  will be examined in detail in section 3. At the free-stream boundary, y = ymax, the ow is assumed to be laminar and irrotational. For the

p

or 4

jw = 2 (d k=dy)2

(18)

d=dyjw = 0:

(19)

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Figure 3: Distribution of (a) the normalized turbulent kinetic energy and (b) its dissipation rate in the fully developed boundary layer on a at plate with zero pressure gradient.

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(b) Figure 2: (a) Skin friction and (b) displacement and momentum thickness distributions in the boundary layer on a at plate with zero pressure gradient.

20

Figure 3 shows the calculated distributions of k and  at a downstream location x = 1:8 m for the dierent boundary conditions for . The results are also compared to the measurements of Karlsson and Johansson [18]. For the turbulent kinetic energy k, we found that the peak near the wall as obtained from the calculations is somewhat lower than that from the measurements, and the computed values are higher than the measurements at the edge of the boundary layer. This is in agreement with the ndings of Sarkar and So [19] for this category of turbulence models. For the dissipation rate , the boundary condition has a signi cant in uence on the  distribution near the wall. Generally, results using equation (19) are closer to the measurements near the wall than those obtained using equation (18). However, the dierent boundary conditions for  have little eect on the near wall k distribution (see gure 3(a)). Also, further away from the wall the results for  from both the ASM and the standard k ?  model agree well with each other. Figure 4 shows the streamwise velocity pro les at the same downstream location as that in gure 3. The results for both models agree very well with the theory [17], although the standard k ?  model results deviate slightly more from the theory in the inner part of the log-law region. Velocity pro les

1/2

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Figure 4: Streamwise velocity pro les in the boundary layer on a at plate with zero pressure gradient using dierent turbulence models. at various locations have been carefully examined, and they all collapse on each other, and agree well with the universal log-law. The second validation case is a turbulent boundary layer subjected to an increasely adverse pressure gradient (both dp=dx and d2 p=dx2 are positive) [20]. For this test case, a grid with 641128 points was used for the calculations. The rst grid point next to the wall varied from y+ = 1:3 to 2.6, depending on the streamwise location. The freestream velocity was 25.5 m=s, while the reference Reynolds number was 1:7  106 =m. In order to match the measurements from Samuel and Joubert [20], the same pressure distribution has to be speci ed at 5

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Zhang, Bachman and Fasel Exp., Samuel & Joubert 1974 Present results

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(b) Figure 5: Distributions of (a) pressure coecient and (b) skin friction coecient on the surface of a

at plate in a turbulent boundary layer ow with an adverse pressure gradient.

T1 T2 T3

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Figure 6: Mean velocity pro les at various streamwise locations. From top to bottom: x = 1:04 m (T1), 1.44 m (T2), 1.79 m (T3), 2.38 m (T4), 2.89 m (T5), 3.39 m (T6).

the freestream boundary of the computational domain. Figure 5(a) shows the pressure coecient along the wall surface from the present calculations, which agrees well with the measurements. The predicted skin friction coecients, using the ASM and the standard k ?  model, respectively, are shown in gure 5(b). They also agree well with the measurements, except near the in ow boundary, where an adjustment had to occur since empirical velocity distributions were imposed as an in ow condition, which do not exactly satisfy the Navier-Stokes equations. In gure 6 the mean velocity pro les from calculations and measurements are compared for various streamwise locations, x = 1:04 m (T1), 1.44 m (T2), 1.79 m (T3), 2.38 m (T4), 2.89 m (T5), 3.39 m (T6). The dotted lines in gure 6 are obtained according to [17]     U = 1 ln yUe U + 5:1 U ; (20) Ue 0:4  Ue Ue where U is the local mean velocity, Ue the local p freestream velocity, and U = 0 = with the wall shear stress 0 from the calculations. For the rst three locations (T1-T3), the pro les are quite similar to those of the boundary layer calculations with zero pressure gradient (see gure 4), indicating that the adverse pressure gradient has not had a large in-

uence on the ow eld at these locations. A large part of the pro les both from the present calculations and from the measurements agrees with the log-law of Clauser [20]. However, due to the adverse pressure gradient the slope of the log-law changes gradually with the location. Starting from T4, the adverse pressure gradient eect sets in strongly, resulting in a distinct wake behavior at the outer part of the pro les. Finally at T6, the pro les not only have the strongest wake behavior, but also deviate strongly from the log-law. Generally, the present calculations with both turbulence models agree well with the measurements [20]. Figure 7 shows the Reynolds shear stress pro les for various streamwise locations, T1{T6. For the rst three locations (T1{T3) in gure 7(a) the pro les are again similar to those with zero pressure gradient, and the predicted results agree well with the measurements [20]. However, further downstream (see curve for T4 in gure 7(b)), as the adverse pressure gradient increases, the maximum value of the Reynolds stress is pushed away from the wall, and the curve also becomes atter. When comparing to the measurements, the present calculations are somehow overshooting the maximum value, but our results are generally bet6

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4 Unsteady RANS Calculations

1.4 T1 T2 T3 ASM k−ε

1.2 1.0

4.1 Simulation parameters

For a boundary layer ow with a laminar in ow and subjected to a strong adverse pressure gradient, as shown in gure 1, a grid of 641128 points was used for all calculations. For an LES, which was performed for comparison, an additional 11 modes were used in the spanwise direction. The grid independency of the solution was checked by using the characteristic parameters of the separation bubble (such as the length and the location of the separation bubble). The incoming ow velocity was 6.65 m=s, and the Reynolds number based on the momentum thickness at the separation point was about 218. The freestream pressure distribution was imposed to match that from the experiments, for which a long separation bubble was found [16]. This case has been taken as a validation test for a 2D \DNS" [22], where relevant parameters of the separation bubble were found to be in an good agreement with the experiments [16] as well as with the other simulations [23]. As for the turbulent, reattached ow, turbulence models were ramped in spatially at the timeaveraged reattachment location of the separation bubble, since it was found in the experiments that a rapid transition occurred at that position [16].

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(b) Figure 7: Reynolds shear stress pro les at various streamwise locations. From top to bottom: (a) x = 1:04 m (T1), 1.44 m (T2), 1.79 m (T3); (b) x = 2:38 m (T4), 2.89 m (T5), 3.39 m (T6).

4.2 2D Unsteady RANS calculations with ASM

First, a 2D Unsteady RANS (URANS) calculation with the ASM was performed. Contours of the instantaneous spanwise vorticity, streamlines, pressure coecient and velocity vectors are plotted in gure 8. Clearly, vortical structures, denoted as A, B and C in gure 8(a), are present in the turbulent boundary layer downstream of the \mean" reattachment location. These structures are gradually dissipated due to the strong turbulent dissipation in the boundary layer. However, the structures are still able to signi cantly reduce the instantaneous streamwise velocity near the wall, as shown in gure 8(d). Upstream of the reattachment point (x=L  10:4), the ow is laminar, with a long separation bubble starting at about x=L = 10:4 (denoted as S in gure 8(a)). The time averaged length of the separation bubble is about 9.4 cm, which agrees well with the experimental measure-

ter than those obtained from the Cebeci and Smith model [21]. At T5 and T6 ( gure 7(b)), the overshoot is increasing, although the locations of the maximum Reynolds stress as obtained from the calculations agree well with the measurements. Also, in general, there is not very much dierence between the results from the ASM and the standard k ?  model. It should be emphasized that no vortical structures in the ow eld could be identi ed in the RANS calculations for both turbulence models, although the calculations were carried out in an unsteady fashion (URANS). 7

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x/L (d) Figure 8: Unsteady RANS calculations using the ASM. Instantaneous contours of spanwise vorticity (a), streamfunction (b), pressure coecient (c) and instantaneous velocity vectors (d).

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ments [16]. The instantaneous turbulence characteristics are shown in gure 9. It can be seen that the turbulent kinetic energy k, turbulent dissipation rate , Reynolds shear stress 12 and \eective eddy viscosity" distributions are clearly associated with the vortical structures in the ow eld (compare gures 9(a)-(d) with gure 8(a)). Close to the \mean" reattachment location, a strong vortical structure, denoted as A in gure 8(a), is shed from the separation bubble. This structure produces strong shear stresses at its edge. The turbulence production and dissipation as well as the shear stress are strongest near the top of the structure, where low momentum

uid is transported by the vortical structure to the outer part of the boundary layer. On the other hand, when high momentum uid is brought back towards the wall by the same vortical structure and stagnates at the at plate surface, another peak of

turbulence production and dissipation appears near the wall (between 14:6  x=L  15:1). Meanwhile, at the center of the shed vortical structure, turbulence production and dissipation are weak due to smaller shear stresses that exist there (see gure 9(c), location A). Farther downstream, a similar pattern can be found for structures B and C. The turbulent kinetic energy and dissipation rate are strong near the top of the structure and at the stagnation point, while remaining weak in the vortex center. The \eective eddy viscosity" shown in gure 9(d) is increasing in the downstream direction, indicating that the contribution from the turbulence model increases in downstream direction. 8

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x/L (e) Figure 10: Same as in gures 8 and 9 except that the standard k ?  model was used. Instantaneous contours of spanwise vorticity (a), turbulent kinetic energy k (b), turbulent dissipation rate  (c), Reynolds shear stress S12 (d) and eddy viscosity (e).

As can be observed in gure 10(a), the boundary layer separates at approximately the same location (x=L = 10:4) as for the previous ASM calculations, which is in agreement with the measurements. The reattachment point of the boundary layer is predicted at x=L = 14:1 and thus a length of the separation bubble of 9.4 cm is obtained. This agrees well with the measurements and the ASM calculations. However, a fundamental dierence between the results from the ASM and from the standard k ?  model is the fact that no noticable unsteady

ow structures were observed for the standard k ?  model. Rather, the ow remained steady throughout, even downstream of the reattachment point (see gure 10(a)-(e)).

4.3 2D Unsteady RANS calculations with standard k ?  model For comparison, a 2D Unsteady RANS (URANS) calculation with the standard k ?  model was performed for the same ow geometry and the same calculation parameters as above. The only change was the turbulence model. Figure 10 shows the instantaneous distributions of spanwise vorticity, turbulent kinetic energy k, turbulent dissipation rate , Reynolds shear stress 12 and eddy viscosity. 9

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4.4 Other simulation results for comparison

and 3D LES with a standard Smagorinsky model were also performed. It was found that the URANS calculations with the standard k ?  model failed to yield any unsteady ow structures at all. In contrast, although predicting the frequency of the large vortical structures reasonable well, the 2D \DNS" yielded results where the structures were too strong, since no explicit turbulent dissipation was provided in these calculations. The results from the ASM calculations for both the frequency and the strength of the large structures were closer to those from the LES with the standard Smagorinsky model.

In order to validate the unsteady results from the ASM, an equivalent calculation was performed for the same ow without any turbulence model (2D \DNS"). The instantaneous spanwise vorticity contours obtained from this calculation are displayed in gure 11. The vortical structures are now stronger than those from the ASM calculations (see gure 8(a)). This is, of course, not surprising since no turbulent dissipation is provided in those calculations. Nevertheless, the frequency of the vortex shedding of the 2D \DNS" results is approximately the same as that of the ASM calculations. In addition, an LES using a standard Smagorinsky model was performed for further comparison. Figure 12(a) shows the instantaneous contours of the zeroth mode of spanwise vorticity (2D component). It is obvious that large 2D organized structures are present in the ow eld, which correspond to the large vortical structures captured in the ASM calculations. This is an indication that the ASM captures the largest structures, while the contribution of all other scales is modeled. In fact, the frequency of the large coherent structures of the LES agrees well with that of the ASM calculations. As for the subgrid scale shear stress 12 and eective eddy viscosity in the LES shown in gures 12(b) and (c), respectively, we found that they are much smaller than the Reynolds shear stress 12 and \effective eddy viscosity" in the ASM calculations (see gures 9(c) and (d), respectively). This is of course consistent with the fact that many of the smaller structures which are captured in the LES do not exist in the ASM calculations. These small structures resolved in the LES contain a large portion of the turbulent energy, which is modeled in the ASM calculations.

Acknowledgement This work was supported by the Oce of Naval Research under contract number N00014-99-1-0885. Dr. Candace Wark and Dr. L. Patrick Purtell served as the technical monitors.

References [1] W. C. Reynolds. The Potential and Limitations of Direct and Large Eddy Simulations. In J. L. Lumley, editor, Whither Turbulence? Turbulence at the Crossroads, Ithaca, N.Y., Mar. 1989. [2] F. Liu and S. H. Ji. Unsteady Flow Calculations with a Multigrid Navier-Stokes Method. AIAA Journal, 34(10):2047{53, Oct. 1996. [3] E. Guilmineau, J. Piquet, and P. Queutey. Unsteady Two-Dimensional Turbulent Viscous Flow past Aerofoils. International Journal for Numerical Methods in Fluids, 25(3):315{66, Aug. 1997.

5 Concluding Remarks

[4] J. A. Ekaterinaris, N. N. Sorensen, and F. Rasmussen. Numerical Investigation of Airfoil Dynamic Stall in Simultaneous Harmonic Oscillatory and Translatory Motion. Transactions of the ASME. Journal of Solar Energy Engineering, 120(1):75{83, Feb. 1998.

In the present study, the Algebraic Stress Model (ASM) was implemented in a fourth-order accurate Navier-Stokes code to simulate an unsteady turbulent boundary layer where large vortical structures are present. Two-dimensional (2D) Unsteady Reynolds-Averaged Navier-Stokes (URANS) calculations with the ASM can capture these structures. This is mainly due to the fact that the \eective eddy viscosity" used in the ASM is strongly coupled with the unsteady ow eld. For comparison, 2D URANS with the standard k ?  model, 2D \DNS",

[5] G. R. Srinivasan, J. A. Ekaterinaris, and W. J. McCroskey. Evaluation of Turbulence Models for Unsteady Flows of an Oscillating Airfoil. Computers & Fluids, 24(7):833{61, Sept. 1995. [6] S. Kumarasamy, R. A. Korpus, and J. B. Barlow. Computation of Noise Due to the Flow 10

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Zhang, Bachman and Fasel

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Over a Circular Cylinder. In Second Computational Aeroacoustics (CAA) Workshop on Benchmark Problems, Maryland Univ., College Park, MD, 1997.

[16] M. Gaster. The Structure and Behavior of Laminar Separation Bubbles. Technical Report R. & M. No. 3597, Aeronautical Research Council, London, Mar. 1967. [17] H. Schlichting. Boundary-Layer Theory. McGraw-Hill, New York, seventh edition, 1979. [18] R. I. Karlsson and T. G. Johansson. LDV Measurements of Higher Order Moments of Velocity Fluctuations in a Turbulent Boundary Layer. In D. F. G. et al. Durao, editor, Laser Anemometry in Fluid Mechanics, pages 273{ 289, Portugal, 1988. Ladoan-Instituto Superior Technico. [19] A. Sarkar and R. M. C. So. A Critical Evaluation of Near-Wall Two-Equation Models against Direct Numerical Simulation Data. Int. J. Heat and Fluid Flow, 18:197{208, 1997. [20] A. E. Samuel and P. N. Joubert. A Boundary Layer Developing in an Increasely Adverse Pressure Gradient. J. Fluid Mech., 66, part 3:481{505, 1974. [21] T. Cebeci and A.M.O. Smith. Analysis of Turbulent Boundary Layers. Academic Press, New York, 1974. [22] H. L. Zhang and H. F. Fasel. Numerical Investigation of the Evolution and Control of TwoDimensional Unsteady Separated Flow over a Stratford Ramp. AIAA Paper 99-1003, Jan. 1999. [23] M. D. Ripley and L. L. Pauley. The unsteady structure of two{dimensional steady laminar separation. Phys. Fluids, 5(12):3099{3106, Dec. 1993.

[7] S. H. Johansson, L. Davidson, and E. Olsson. Numerical Simulation of Vortex Shedding past Triangular Cylinders at High Reynolds Number Using a k ?  Turbulence Model. International Journal for Numerical Methods in Fluids, 16(10):859{78, May 1993. [8] A. Pentaris and S. Tsangaris. Numerical Simulation of Unsteady Viscous Flows Using an Implicit Projection Method. International Journal for Numerical Methods in Fluids, 23(9):897{921, Nov. 1996. [9] R. E. Neel, R. W. Walters, and R. L. Simpson. Computations of Steady and Unsteady Low-Speed Turbulent Separated Flows. AIAA Journal, 36(7):1208{15, July 1998. [10] T. B. Gatski and C. G. Speziale. On Explicit Algebraic Stress Models for Complex Turbulent Flows. J. Fluid Mech., 254:59{78, Sept. 1993. [11] H. M. Meitz and H. F. Fasel. A CompactDierence Scheme for the Navier-Stokes Equations in Vorticity-Velocity Formulation. J. Computational Phys., accepted for publication, 1999. [12] W. P. Jones and B. E. Launder. The Prediction of Laminarization with a Two-Equation Model of Turbulence. Inter. J. of Heat and Mass Transfer, 15:301{314, 1972. [13] B. E. Launder and B. I. Sharma. Application of the Energy Dissipation Model of Turbulence to the Calculation of Flow near a Spinning Disc. Letters in Heat and Mass Transfer, 1(2):131{ 138, Feb. 1974. [14] J. Smagorinsky. General Circulation Experiments with the Primitive Equations. I. The Basic Experiment. Mon. Weather Rev., 91:99{ 164, 1963. [15] M. Kloker and H. F. Fasel. Direct Numerical Simulations of Boundary Layer Transition under Strong Adverse Pressure Gradient. In Proc. IUTAM Symp. Laminar-Turbulent Transition, Sendai, Japan, 1993. 11

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