A Hybrid RANS LES Simulation of Turbulent Channel Flow

8/8/2019 A Hybrid RANS LES Simulation of Turbulent Channel Flow

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Digital Object Identier (DOI) 10.1007/s00162-003-0089-xTheoret. Comput. Fluid Dynamics (2003) 16: 387403 TheoreticalandComputational

Fluid Dynamics

A Hybrid RANS / LES Simulationof Turbulent Channel Flow

Fujihiro Hamba

Institute of Industrial Science, University of Tokyo,Komaba, Meguro-ku, Tokyo 153-8505, Japan

[email protected]

Communicated by M.Y. Hussaini

Received 5 April 2002 and accepted 8 January 2003Published online 25 March 2003 Springer-Verlag 2003

Abstract. Hybrid models combining large eddy simulation (LES) with Reynolds-averaged NavierStokes(RANS) simulation are expected to be useful for wall modeling in the LES of high Reynolds number ows.Some hybrid simulations of turbulent channel ow have a common defect; the mean velocity prole hasa mismatch between the RANS and LES regions due to a steep velocity gradient at the interface. This mis-match is reproduced and examined using a simple hybrid model; the Smagorinsky model is switched toa RANS model increasing the lter width. It is suggested that a rapid spatial variation in the eddy viscos-ity is responsible for an underestimate of the grid-scale shear stress and for the steep velocity gradient. Toreduce the mean velocity mismatch a new scheme is proposed; additional ltering is introduced to denetwo kinds of velocity components at the interface between the two regions. The two components are usedto remove inconsistency in the velocity equations due to a rapid variation in the lter width. Using the newscheme, simulations of channel ow are carried out with the simple hybrid model. It is shown that the grid-scale shear stress becomes large enough and most of the mean velocity mismatch is removed. Simulationsfor higher Reynolds numbers are carried out with the k model and the one-equation subgrid-scale model.Although it is necessary to improve the turbulence models and the treatment of the buffer region, the newscheme is shown to be effective for reducing the mismatch and to be useful for developing better hybridsimulations.

1. Introduction

In large eddy simulation (LES) physical quantities such as the velocity are decomposed into a resolved grid-scale (GS) component and an unresolved subgrid-scale (SGS) component. For an accurate simulation of turbulent ow grid spacing needs to be small enough compared with the turbulent integral scale. The com-puting cost of LES is much larger than that of Reynolds-averaged NavierStokes (RANS) simulation ingeneral. LES was rst used to compute basic ows in simple geometries for turbulence research. Due to therapid development of computers it can now be applied to practical engineering ows in relatively complexgeometries. However, it is still impossible to simulate wall-bounded ows at very high Reynolds numberswith the no-slip boundary conditions. This is because many grid points are required to resolve small turbulentstructures near the wall. For example, Chapman (1979) estimated that the number of grid points needed to re-solve the inner boundary layer is proportional to Re 1.8 . According to an estimate by Spalart et al. (1997) the

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388 F. Hamba

number of grid points is of the order of 10 11 for the LES of an airplane wing. For the atmospheric boundarylayer the Reynolds number is much higher compared with engineering ows.

To alleviate the near-wall resolution requirement, approximate wall boundary conditions are often im-posed; they represent the effects of the unresolved near-wall region on the outer layer (Deardorff, 1970;Schumann, 1975; Moeng, 1984; Piomelli et al. , 1989). Wall shear stress at the rst off-wall grid point isevaluated assuming a wall law of the velocity and a simple relationship between the velocity and the wallstress. This approach has met limited success; the approximate boundary conditions are not accurate enoughfor high Reynolds number ows (Nicoud et al. , 2001).

To improve wall modeling, two-layer approximate boundary conditions were developed (Balaras et al. ,1996; Cabot, 1995). In this approach a ne grid is embedded in the rst off-wall cell in a coarse grid. Tur-bulent boundary layer equations are solved on the embedded grid to estimate the wall stress. Balaras et al.(1996) successfully simulated ows in a plane channel, a square duct, and a rotating channel. This model-ing was also applied to ows behind a backward facing step by Cabot and Moin (1999) and to ows pasta trailing edge by Wang and Moin (2001). Another approach to wall modeling is to use data from directnumerical simulation (DNS) or accurate LES in order to nd a model that reproduces the wall stress corres-ponding to a given velocity eld. Using data from the DNS of channel ow at the friction Reynolds numberof Re =180 Bagwell et al. (1993) obtained the conditional average of the wall stress given a velocityeld by linear stochastic estimation. Recently, using suboptimal control theory and adopting the logarith-mic law as a target velocity prole Nicoud et al. (2001) successfully carried out the LES of channel ow atRe =4000. They used the LES results as reference data for the linear stochastic estimation to derive a betterwall model for high Reynolds number ows.

In the approaches mentioned above the rst off-wall grid point is often located in the logarithmic region.Since the grid width in all three directions is larger than the local turbulent integral scale, approximate wallboundary conditions are necessary. An alternative approach is to rene the grid in the wall-normal directionwhile keeping it broad in the wall-parallel directions; this approach can also alleviate the near-wall reso-lution requirement. If the steep wall-normal gradients of boundary layers are captured, the no-slip boundaryconditions can be used. However, the wall-parallel grid is still larger than the local integral scale. Since thenear-wall small structures are not resolved, conventional SGS models cannot be used. This type of grid iscommonly used in the RANS of the turbulent boundary layer. Therefore, a simple idea to improve the SGSmodel near the wall is to adopt a RANS eddy viscosity model (Schumann, 1975; Sullivan et al. , 1994).Baggett (1998) made an attempt to use the v2 f RANS model of Durbin (1995) to supplement the SGS modelin the near-wall region. Spalart et al. (1997) proposed the detached eddy simulation by modifying the one-equation SpalartAllmaras RANS model. Changing the model length scale from the wall distance to the gridwidth, they switched the RANS simulation in the near-wall region to the LES in the outer region. Nikitinet al. (2000) compared several detached eddy simulations of turbulent channel ow; they were able to treatows at Re =80 000. Several massively separated ows have been calculated by the detached eddy sim-ulations (Strelets, 2001). Some of the simulations are based on the shear stress transport model of Menter(1994). Davidson and Peng (2001) carried out similar hybrid simulations of ows in a channel and overa two-dimensional hill. They adopted the k RANS model near the wall and the one-equation SGS modelfor LES away from the wall.

Hybrid LES / RANS approaches are expected to be useful not only for wall modeling but also for otherpurposes. An example is the inow boundary conditions for LES (Batten et al. , 2001). In LES, approxi-

mate inow boundary conditions are necessary; an instantaneous velocity eld should be generated by somemeans. When a RANS model is used in the upstream region, only the averaged quantities are necessary asthe boundary conditions. If at the interface between the upstream and downstream regions an adequate GSkinetic energy is automatically transferred from the kinetic energy in RANS, this hybridsimulation can be analternative inow condition for LES. Contrary to wall modeling explained above, Hamba (2001) carried outa hybrid simulation of channel ow with the SGS model near the wall and the k model away from the wall.In this simulation, LES can be considered a kind of boundary condition for RANS. Such a hybrid simulationis useful for RANS in which unsteady properties are essential at the boundary. For example, when the soundwave intensity is calculated by a RANS-type model, unsteady uid motions near the body surface need to besolved by LES. To combine LES with RANS is also very interesting from the theoretical point of view. TheRANS equation is based on ensemble averaging. If the turbulent ow is statistically homogeneous in somedirections, this average is expected to be equivalent to the spatial average. On the other hand, ltering in LES


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A Hybrid RANS / LES Simulation of Turbulent Channel Flow 389

corresponds to averaging over a local volume. It is not clear whether LES can be switched to RANS only byincreasing the lter width.

Turbulent channel ow is a good test case for wall modeling and hybrid simulations. Although hybridsimulations have met some success, some simulations of channel ow show a common defect in predictingthe mean velocity prole. In detached eddy simulations by Nikitin et al. (2000) the mean velocity proleshave mismatches between the RANS and LES regions. The velocity gradients are too steep near the inter-face; the resulting skin-friction coefcients are about 15% lower than expected values. The mean velocityproles obtained by Davidson and Peng (2001) also show a kink at the matching line. In the hybrid simu-lation by Hamba (2001) a similar mismatch is seen in the mean velocity prole although the positions of RANS and LES are reversed compared with the other two simulations. There seems to be some commoncause of the mismatch; it does not depend on details of the turbulence models. In this work, in order to inves-tigate the cause of the mismatch and to improve the mean velocity prole, we carry out hybrid simulationsof channel ow.

In the following section using a simple hybrid model we reproduce and examine the mean velocity mis-match between the two regions in the channel ow. In Section 3 we propose a new scheme to reduce themismatch and apply it to the simulation with the simple hybrid model. In Section 4 using the new schemewe carry out hybrid simulations with the k model and the one-equation SGS model. Concluding remarks

are given in Section 5.

2. Mean Velocity Mismatch

As mentioned in the previous section the mean velocity proles predicted by hybrid simulations of channelow have mismatches between the RANS and LES regions. Nikitin et al. (2000) carried out detached eddysimulations with three independent codes; they compared eight runs at different Reynolds numbers. All re-sults except for the lowest Reynolds number case show similar mismatches. One of them is plotted as thedashed line in Figure 1. At 30 < y+ < 200, where y+ is the wall-normal coordinate in wall unit, the meanvelocity prole in the RANS region agrees well with the logarithmic law plotted by the solid line. However,at the bottom of the LES region, 200 < y+< 700, the velocity gradient is too steep; the velocity prole at y+ > 700 rises above the logarithmic law. Since the ow rate is overestimated, the skin-friction coefcientis 11% lower than the expected value in this case. The dotted line denotes the result from a hybrid simu-lation by Davidson and Peng (2001); the matching line is located at y+=60. At the bottom of the LESregion, 60 < y+< 100, the velocity gradient is too steep. The dot-dashed line denotes the result from a hy-brid simulation by Hamba (2001); a similar mismatch is clearly seen near y+=300. Turbulence models andmethods of combining them are different between the three simulations. Nevertheless, the velocity proleshave similar mismatches between the RANS and LES regions.

Figure 1. Mean velocity proles obtained from three hybrid simulations of channel ow. The solid line denotes the logarithmic law.


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Table 1. Filter width and the location of buffer region for six runs.

Case at 0 < y+ < y+ A at y+ B < y+ < 590 y+ A y+ B1 h f s h 210 2102 210 210

3 h 210 2104 h f s 210 2105 h 161 2386 h 122 269

Figure 2. Mean velocity proles for Cases 1 and 2. The solid line denotes the result from DNS by Moser et al . (1999).

The periodic boundary conditions are used in the x and z directions whereas the no-slip conditions are im-posed at the wall ( y =0 and y =2). We use the second-order nite-difference scheme in space and theAdamsBashforth method for time marching. The computation was run for a sufciently long time to be sta-tistically independent of the initial condition; then statistics such as the mean velocity were accumulated overa time period of 20.

In Case 1 the grid width is used as the lter width. The simulation is a typical LES of channel ow; an un-steady three-dimensional velocity eld is obtained. On the other hand, in Case 2 the lter width is set equalto the integral scale throughout the domain. The resulting eddy viscosity is so large that the uctuation u ivanishes and a steady one-dimensional solution u i ( x, y, z, t ) = u i ( y) is obtained. This simulation can beconsidered a RANS. Figure 2 shows the mean velocity proles u for Cases 1 and 2. The dashed line de-notes the prole in Case 1 and the dotted line stands for that in Case 2; both proles agree well with the DNSdata plotted by the solid line. This result shows that the eddy viscosity with obtained from (6) acts as an ad-equate mixing length model. Therefore, it can be used as a RANS model in hybrid simulations in Cases 36.In Cases 3, 5, and 6 the RANS model is solved near the wall whereas LES is carried out away from thewall. These cases correspond to the wall-modeling approaches by Nikitin et al. (2000) and by Davidson andPeng (2001). In Case 4 the positions of RANS and LES are reversed; they are the same as Hamba (2001).

Figure 3(a) shows the mean velocity prole for Case 3. The vertical line at y+=210 indicates the locationwhere the lter width is switched from to h . A mismatch is clearly seen; the velocity gradient is too steepat the bottom of the LES region. Figure 3(b) shows the proles of the shear stresses for Case 3; the GS partdenotes u v , the SGS part u v , and the viscous part u / y. In the LES region at y+ > 210 theGS part dominates as a typical LES. In the RANS region at y+< 210 the GS part decreases to very smallvalues and the SGS part takes over. We should note that the nonzero value of the GS part in the RANS regionimplies temporal and spatial uctuations of the mean velocity u i . In this case the simulation in the RANS re-gion can be considered an unsteady RANS. Figure 3(c) shows the prole of the mean eddy viscosity t forCase 3. In the RANS region the eddy viscosity is very large; its maximum value is about 50 times as largeas the molecular viscosity . At the interface the eddy viscosity drops sharply; it is comparable with in theLES region. A similar prole of the eddy viscosity was also reported in Davidson and Peng (2001). Figure 4shows the proles of the mean velocity and the shear stresses for Case 4. A similar steep velocity gradient is


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392 F. Hamba

Figure 3. Proles of mean velocity, shear stresses, and mean eddy viscosity for Case 3: (a) u , (b) u v , u v +u v , andu v +u v u / y, and (c) t . The solid line for mean velocity denotes the result from DNS by Moser et al . (1999).

Figure 4. Proles of mean velocity and shear stresses for Case 4: (a) u and (b) u v , u v +u v , and u v +u v u / y. The solid line for mean velocity denotes the result from DNS by Moser et al . (1999).

seen near the interface although the position of the RANS and LES regions is different from that in Case 3.The dominance of the GS shear stress in the LES region and that of the SGS part in the RANS region aresimilar to those in Case 3. Therefore, the mean velocity mismatch is related not only to wall modeling butalso to general hybrid simulations.

Following the analysis by Nicoud et al. (2001) we examine the relationship between the velocity gradi-ent and the shear stress in Case 3. Considering the balance between the convection term and the pressuregradient in the mean velocity equation we can see that the sum of the three shear stresses, u v +u v u / y, is equal to y1 as shown in Figure 3(b). Assuming that the SGS shear stress can be approximated


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by

u v = t u y +

v x = t

u y

, (7)

we have u y =

u v +1 y + t

. (8)

At y =0.4 the mean eddy viscosity t is comparable with in Figure 3(c); this small value is adequate be-cause this location lies in the LES region. However, the magnitude of the GS shear stress u v is not largeenough at y =0.4 in Figure 3(b). This leads to a large value of the numerator on the right-hand side in (8), thedifference between the GS and total stresses. Therefore, an underestimate of the GS shear stress at the bot-tom of the LES region is responsible for the steep velocity gradient. One may expect that if the denominatoron the right-hand side in (8) is large enough at y =0.4 to balance the numerator, then the velocity gradientdecreases to a normal value. A simple idea to increase the eddy viscosity t is to set a larger value of con-stant C s in (4) only at y =0.4. However, this approach is not effective in reducing the mismatch; the reasonis explained as follows. The eddy viscosity is a function of the product of C s and in (4); the increase of C skeeping unchanged is equivalent to the increase of keeping C s unchanged. This means that the RANSregion is effectively extended to y =0.4 where is increased; the interface between the RANS and LES re-gions is shifted a grid point away from the wall. A steep velocity gradient still exists near the new interface.Therefore, it is not enough just to increase C s in order to decrease the velocity gradient.

Here, in order to examine why the GS shear stress is underestimated near the interface, we considernonlocal properties of turbulent eld (Hamba, 1995). Due to the effects of the pressure and the turbulent dif-fusion, the time development of the GS velocity at one point can be affected by the velocity at other pointswithin a distance of the integral scale. For example, in Case 1 the GS velocity u i at y =0.4 can be inuencedby that around y =0.3. However, in Case 3 the GS uctuation at y =0.3 in the RANS region is very small; itcannot account for the nonlocal effects of the uctuations on the GS velocity at y =0.4. Although the SGSmodel should represent such an effect, the usual eddy viscosity model does not seem to act well due to itsdissipative characteristics.

In the three previous hybrid simulations the same equations (1)(3) are used. Although the method of evaluating the eddy viscosity t is different, it is common that the value of t changes from a large RANSvalue to a small SGS value. In the simulations by Davidson and Peng (2001) and by Hamba (2001) the eddyviscosity is switched at one point like Case 3 in the present analysis. On the other hand, in the detachededdy simulation by Nikitin et al. (2000) it is varied smoothly in the buffer region. The result of Case 3 doesnot directly explain the velocity mismatch in the detached eddy simulation. To examine cases in which theeddy viscosity varies smoothly, we set the buffer region in Cases 5 and 6. Three and six cells are located inthe buffer region at y+ A < y+ < y+ B in Cases 5 and 6, respectively. The prole of the lter width is plottedin Figure 5(a). The resulting eddy viscosity t also changes smoothly compared with Case 3 as shown inFigure 5(b). Figure 5(c) shows the mean velocity proles in Cases 3, 5, and 6. In Cases 5 and 6 the velocitygradient actually decreases in the buffer region compared with that in Case 3. However, since the buffer re-gion becomes wide and the steep velocity gradient is accumulated, the mean velocity in the LES region does

not decrease in Cases 5 and 6. The center-line velocity is nearly the same in the three cases. Varying theeddy viscosity smoothly does not reduce the velocity mismatch. This result suggests that the cause of themismatch shown in Nikitin et al. (2000) is similar to that in the other two simulations.

3. New Scheme by Additional Filtering

In the previous section we examined the cause of the velocity mismatch. In the RANS region the eddy vis-cosity is large and the uctuations of GS velocity are very small. Since the nonlocal contribution from theuctuations in the RANS region is reduced, the GS shear stress is not large enough near the interface in theLES region. The underestimate of the GS shear stress leads to the steep velocity gradient. This result sug-gests that by changing only the value of the eddy viscosity from a large RANS value to a small SGS value we


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394 F. Hamba

Figure 5. Proles of lter width, mean eddy viscosity, and mean velocity for Cases 3, 5, and 6. The solid line for mean velocitydenotes the result from DNS by Moser et al . (1999).

Figure 6. Grid cells and velocity components dened in (a) the old scheme and (b) the new scheme.

cannot avoid the velocity mismatch. The rapid change in the eddy viscosity corresponds to that in the lterwidth in the LES framework. Such a rapid change is not seen in usual LES; it may cause some inconsistencyin hybrid simulations. Here, we examine the inconsistency paying attention to numerical schemes.

In this work the second-order nite-difference scheme is used in a three-dimensional staggered grid.Figure 6(a) shows grid cells and velocity components near the interface between the RANS and LES regions.Only the x y plane is shown for simplicity. Grid cells (i , j) and (i , j +1) are located in the RANS and LESregions, respectively, and grid line j +12 denotes the interface. For an explanation it is assumed that the lter


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width is set to 5 h and h for cells in the RANS and LES regions, respectively, and that is effectively equalto 3h at the interface. Superscripts attached to overbars denote the lter width. For example, the discretizedforms of the continuity equation for cells (i, j +1) and (i , j) are given by

u h

x +

wh

z i, j+1 +vhi, j

+3/ 2

v3hi, j

+1/ 2

h y =0, (9) u5h

x + w5h

zi, j

+v3hi, j+1/ 2 v

5hi, j1/ 2

h y =0, (10)

respectively, where discretized forms of the x- and z-derivatives are omitted and h y denotes the grid widthin the y direction. The velocity component v3hi, j+1/ 2 included in both (9) and (10) represents the mass uxthrough the interface between the two cells. This component is associated with =3h ; this means the massux averaged in the line at xi3/ 2 < x < xi+3/ 2 in the sense of the control volume method. It is not consistentwith the other components with =h in (9). For the continuity equation (9) to be consistent, v3hi, j+1/ 2 shouldbe replaced by

vhi, j

+1/ 2 . However, (10) cannot be consistent at the same time because the other components

are associated with =5h . Similarly, the discretized forms of the streamwise velocity equation for pointsi +12 , j +1 and i +12 , j are described as uhi+1/ 2, j+1

t = 1

h y [( uhv

h ) i+1/ 2, j+3/ 2 ( u 3h v3h ) i+1/ 2, j+1/ 2]+ , (11)

u5hi+1/ 2, jt =

1h y [( u

3h v3h ) i+1/ 2, j+1/ 2 ( u 5h v5h ) i+1/ 2, j1/ 2]+ , (12)respectively, where only the convection terms in the y direction are shown explicitly on the right-hand sides.At node i +12 , j +12 the velocity components are not dened. The convection term is evaluated by theinterpolation

u3h v3h i+1/ 2, j+1/ 2 =u hi+1/ 2, j+1 +u

5hi+1/ 2, j v

3hi+1, j+1/ 2 +v

3hi, j+1/ 2

4. (13)

The term u3h v3h represents the momentum ux through the interface; it should be replaced by uh vh in (11)like the mass ux in (9). In general, spatial and temporal uctuations of velocity decrease as the lter widthincreases; the magnitude of u 3h v3h is less than that of u h vh on average. This means that the uctuation of uhi+1/ 2, j+1 driven by the convection through the interface is underestimated. This underestimate must be re-sponsible for the small value of the GS stress at y =0.4 in Figure 3(b). The inconsistency is related to thefact that the ltering operation does not commute with the spatial derivative in a nonuniform grid. Althougha nonuniform grid is often used in LES, variation in grid spacing is not very large and the commutationerror is small (Fureby and Tabor, 1997). Since the lter width varies so rapidly near the interface in hybrid

simulations, the commutation error cannot be neglected.In order to remove this inconsistency we introduce additional ltering to dene two kinds of velocitycomponents at the interface between the RANS and LES regions. Figure 6(b) shows the grid cells and vel-ocity components used in the new scheme. At point i, j +12 two components vh and v5h are dened for thewall-normal velocity. The continuity equation can now be given by

u h x +

wh z i, j+1 +

vhi, j+3/ 2 vhi, j+1/ 2

h y =0, (14)

u5h x +

w5h z

i, j+

v5hi, j+1/ 2 v5hi, j1/ 2

h y =0 (15)


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396 F. Hamba

for the two cells. The value of vhi, j+1/ 2 in (14) is obtained by solving its transport equation whereas that of v5hi, j+1/ 2 in (15) is given by the following relationship:

v5hi, j+1/ 2 =vh i, j+1/ 2 , (16)where denotes additional ltering in the x and z directions; its width is =2 6h = (5h )2 h 2 inthis case. The component vhi, j+1/ 2 represents the GS mass ux averaged over =h whereas v

5hi, j+1/ 2 repre-sents that averaged over =5h . The former ux comes into cell ( i, j +1) and the latter ux goes out of cell(i , j) when the value is positive. The continuity condition holds due to the relationship (16). Introducing the

two components we can consistently evaluate the continuity equation at both cells. Similarly, the convectionterms in the streamwise velocity equation are given by

u hi+1/ 2, j+1t =

1h y u

h vh i+1/ 2, j+3/ 2 uh vh i+1/ 2, j+1/ 2 + , (17)

u 5hi

+1/ 2, j

t = 1

h y uh

vh

i+1/ 2, j+1/ 2 u5h

v5h

i+1/ 2, j1/ 2 + , (18)where

u h vh i+1/ 2, j+1/ 2 =uhi+1/ 2, j+1 +u

5hi+1/ 2, j v

hi+1, j+1/ 2 +v

hi, j+1/ 2

4. (19)

Like the mass ux v5hi, j+1/ 2 in (15) the momentum ux uh vh i+1/ 2, j+1/ 2 in (18) is obtained by additionalltering with . In (17) the momentum ux u h vh i+1/ 2, j+1/ 2 is adopted instead of u

3h v3h i+1/ 2, j+1/ 2in (11); we expect that the uctuation of u hi+1/ 2, j+1 is evaluated adequately and therefore the velocity gra-dient is reduced. To be more accurate, u 5hi

+1/ 2, j in (19) should be replaced by uhi

+1/ 2, j . Since the transport

equation for u 5hi+1/ 2, j is solved at point i +12 , j in the RANS region, deltering is needed to obtain u hi+1/ 2, jfrom u 5hi+1/ 2, j . However, it was found that a straightforward application of deltering is ill-posed for sucha large ratio of the lter widths. We use u5hi+1/ 2, j as it stands although some approximation to the replacementshould be employed in future.

The transport equation for the spanwise component w is modied in a similar way to (17)(19). Sincethe discretized forms of the continuity equation are modied, those of the Poisson equation for the pres-sure should also be changed, introducing two kinds of the pressure. Additional modications including thePoisson equation are explained in Appendix.

To assess the validity of the new scheme we carry out two more runs as shown in Table 2. In Cases 3N and5N the new scheme is applied to simulations; other conditions are the same as Cases 3 and 5, respectively.In Case 3N the lter width is switched at the interface at y+22+1/ 2 =210. The width of the additional lteringis given by

22+1/ 2 = 222 223 =3 11h 23 , (20)

Table 2. Scheme and the location of buffer region for four runs.

Case Scheme y+ A y+ B3 old 210 210

3N new 210 2105 old 161 238

5N new 161 238


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Figure 7. Proles of mean velocity and shear stresses for runs with the new scheme: (a) u for Cases 3N and 5N and (b) u v ,u v +u v , and u v +u v u / y for Case 3N. The solid line for mean velocity denotes the result from DNS by Moseret al . (1999).

where 22 =10h 23 , 23 =h 23 . (21)

In Case 5N three cells are located in the buffer region at y+ A < y+< y+ B . The additional ltering is applied at y+ j+1/ 2 ( j =20 , . . . , 23) successively with

j+1/ 2 = 2 j 2 j+1 , (22)where

20 =16h 20 , 21 =12h 21 , 22 =8h22 , 23 =4h 23 , 24 =h24 . (23)Figure 7(a) shows the mean velocity proles for Cases 3N and 5N. In both cases the velocity gradient nearthe interface decreases compared with the cases using the old scheme. Due to the buffer region the velocityprole in Case 5N is closer to the logarithmic prole than that in Case 3N. This result shows that the newscheme is useful for reducing the velocity mismatch. Figure 7(b) shows the shear stresses in Case 3N. As isexpected, the GS shear stress at y =0.4 is greater than that in Case 3 shown in Figure 3(b). This is why thevelocity gradient decreases at this point in Figure 7(a). It is interesting to point out that the GS shear stressat y =0.3 in the RANS region decreases compared with Case 3. This small uctuation is appropriate as thevelocity u i in the RANS model. It should be noted that the sum of the three stresses in Case 3N is equal to y1 like in Case 3. This is not trivial because two different convection terms, u h vh and u h vh , are dened atthe same point in the buffer region as described in (17) and (18). Such a straight line is obtained because of the relationship u h vh =uh vh . If the equality does not hold, then the prole is discontinuous at the point.

4. Simulation with the k Model and the One-Equation SGS Model

In the previous sections we used a mixing length RANS model to examine the mismatch between theRANS and LES regions. Since the value of the turbulent integral scale is valid only for a channel owat Re =590, the model is not universal. More general models are necessary for actual applications. Here,we adopt the standard k model for RANS (Launder and Spalding, 1974). For LES we use a one-equationSGS model involving the transport equation for the SGS kinetic energy (Yoshizawa and Horiuti, 1985). Theone-equation SGS model is expected to be more adequate than the Smagorinsky model because the kineticenergy equation can be shared in LES and RANS.

In both the LES and RANS regions, (1)(3) for the GS velocity are solved; overbars are interpreted asthe Reynolds average in the RANS region. The eddy viscosity involved in (3) is modeled in terms of the


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398 F. Hamba

turbulent kinetic energy k and its dissipation rate as

t =C f k 2

, (24)

where f is the damping function near the wall,

f =1 exp y+ A

, (25)

and model constants are set to C =0.09 and A =25. The transport equation for k is written as Dk Dt = u i u j

u i x j +

xi

t k +

k xi

, (26)

where k =1. The boundary condition k =0 is imposed at the wall. The equation for the kinetic energy iscommon in RANS and LES whereas the treatment of the dissipation rate is different. In the k model thetransport equation for the dissipation rate for RANS, R , is solved:

DR Dt = C 1

Rk

u i u j u i x j C 2 f

2Rk +

xi

t +

R xi

, (27)

where f is the damping function

f =1 exp y+ A

. (28)

The boundary condition at the wall is R = 2k / y2 . Model constants are set to C 1 =1.44, C 2 =1.92, =1.3, and A =9.5. On the other hand, in the one-equation SGS model the dissipation rate for LES, L ,is algebraically expressed in terms of k and as

L =k 3/ 2

C , (29)

where C =0.61. The one-equation SGS model of this type was shown to give results almost the same asthe Smagorinsky model for channel ows and mixing layers (Horiuti, 1985; Hamba, 1987).The k model is used near the wall whereas the one-equation SGS model is solved away from the wall.

To connect the RANS and LES regions smoothly we set a buffer region at y+ A < y+< y+ B . In this region thedissipation rate is obtained from the interpolation

= y+ B y+ y+ B y+ A

R + y+ y+ A y+ B y+ A

L . (30)

The transport equation for R is modied as

DR Dt = C 1 k u i u j u

i

x j C 2 f R

k + xi t +

R xi

. (31)

From the theoretical point of view it is better to use the same equation for the RANS region and the bufferregion. However, we found that it causes an overestimate of the dissipation rate in the buffer region. In thiswork we adopt (31) and obtain better values of the dissipation rate. In order to use (27) we should adjust thevalue of coefcients C 1 and C 2 in the buffer region; more work needs to be done to improve the treatmentof the dissipation rate in the buffer region. Here, we mention the value of included in (29). In the LESregion at y+y+ B it is set to =h . As y+decreases in the buffer region, the SGS kinetic energy k increaseslike the SGS shear stress shown in Figure 3(b). If the lter width is set to =h in the buffer region, then thedissipation rate L is overestimated. We need to set a value larger than h to balance the increase in k in thebuffer region. This large value is related to the additional ltering introduced in the new scheme.


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Table 3. Parameters for six runs.

Case Re N y Scheme y+ A y+ B7 590 64 old 161 238

7N 590 64 new 161 2388 1180 72 old 318 470

8N 1180 72 new 318 4709 5900 94 old 958 1140

9N 5900 94 new 958 1140

Figure 8. Proles of mean velocity and shear stresses for Case 7: (a) u and (b) u v , u v +u v , and u v +u v u / y. The solid line for mean velocity denotes the result from DNS by Moser et al . (1999).

Using the turbulence models explained above we carry out six runs as shown in Table 3. The new schemewith the additional ltering is adopted in Cases 7N, 8N, and 9N whereas the old scheme is used in Cases 7,8, and 9. In Cases 7 and 7N the Reynolds number Re

=590 is the same as the runs discussed in the previous

sections. Since the k model is used, the prole of the turbulence length scale prescribed for this Reynoldsnumber is not necessary; the Reynolds number can be increased as shown in Cases 8, 8N, 9, and 9N. In all sixcases three cells are placed in the buffer region at y+ A < y+< y+ B ; the location in each case is given in Table 3.In Cases 7N, 8N, and 9N the additional ltering is applied using the lter width similar to (22) and (23).

Figure 8 shows the proles of the mean velocity and the shear stresses for Case 7. Two vertical linesdenote the location of y+ A and y+ B . Since the old scheme is used in Case 7 a mismatch is seen in the meanvelocity prole in Figure 8(a). This velocity prole is similar to those in Cases 5 and 6 in Figure 5(c). In Fig-ure 8(b) the magnitude of the GS shear stress at y =0.46 is not large enough although it is somewhat largerthan that at y =0.4 in Case 3 shown in Figure 3(b). Velocity mismatch appears for both runs with the mixinglength model and with the k model. This result suggests that the mismatch does not depend on the detailsof the turbulence model.

Figure 9 shows the proles of the mean velocity and the shear stresses for Case 7N. Although the mean

velocity obtained from the new scheme is still greater than the DNS result, the steep velocity gradient near y+=y+ B disappears in Figure 9(a). The value of the mean velocity at y+=590 decreases; it is closer to theDNS result compared with that in Case 7. In Figure 9(b) the magnitude of the GS shear stress at y+=y+ Bin Case 7N is large enough. This large value means that the GS velocity uctuations near the interface arerecovered using the new scheme. Near y+=y+ A the velocity gradient is slightly greater than the DNS result.This steep gradient does not seem to be directly related to the connection of the two models. There seems tobe a problem only with the RANS model or with the treatment of the buffer region because the SGS stress isgreater than the GS stress in the regions. In fact, decreasing the model constant C in (24) near y+=y+ A weobtain a velocity prole with a less steep gradient (not shown here).

Figure 10 shows the mean velocity proles for higher Reynolds number cases: Re =1180 for Cases 8and 8N whereas Re =5900 for Cases 9 and 9N. In Cases 8N and 9N the steep velocity gradient near y+=y+ B disappears and the center-line velocity decreases. This result shows that the new scheme is effect-


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400 F. Hamba

Figure 9. Proles of mean velocity and shear stresses for Case 7N: (a) u and (b) u v , u v +u v , and u v +u v u / y. The solid line for mean velocity denotes the result from DNS by Moser et al . (1999).

Figure 10. Proles of mean velocity for higher Reynolds number cases: (a)

u for Cases 8 and 8N and (b)

u for Cases 9 and 9N.

Figure 11. Proles of turbulent kinetic energies for Case 7N: u2i / 2, k , and u

2i / 2 + k . The solid line denotes the result from

DNS by Moser et al . (1999).

ive for higher Reynolds numbers. However, the value of the mean velocity in the LES region at y+> y+ Bis somewhat greater than the logarithmic law. This is due to the steep gradient near y+=y+ A . As in Case7N the steep gradient is caused by the RANS modeling at y+< y+ A or the treatment of the buffer region at y+ A < y+< y+ B . Improvement of the velocity prole in this region remains as future work.


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Figure 11 shows the prole of the turbulent kinetic energies for Case 7N: the GS part u2

i / 2, the SGSpart k , the sum of the two parts, and the DNS result u 2i / 2. The SGS part dominates in the RANS re-gion near the wall whereas the GS part dominates in the LES region away from the wall. The sum of theGS and SGS parts at y < 0.1 is less than the DNS result. This is because simple damping functions are usedin the present k model. To obtain better agreement in this region we need to use a low Reynolds numberk model. Near the buffer region the sum of the two parts shows an oscillatory prole; it increases onceat y =0.33 and then drops sharply at y =0.37. The turbulent kinetic energy is not connected smoothly, un-like the mean velocity. This result also implies that the turbulence model should be improved further. Forexample, the transport equation for the dissipation rate in (31) and a method of changing the lter width in(23) need to be examined.

5. Conclusions

In previous hybrid simulations of channel ow the mean velocity prole has a mismatch between the RANSand LES regions. In this work this mismatch is reproduced and examined using a simple hybrid model; thelter width in the Smagorinsky model is changed from the grid width in the LES region to the turbulent inte-

gral scale in the RANS region. It is suggested that a rapid spatial variation in the eddy viscosity is responsiblefor an underestimate of the GS velocity uctuations and for the steep velocity gradient. To reduce the meanvelocitymismatch a new scheme is proposed; additional ltering is introduced to dene two kinds of velocitycomponents at the interface between the two regions. The two components are used to remove inconsistencyin the continuity equation and the GS velocity equation. Using this scheme, simulations of channel ow arecarried out with the simple hybrid model. As a result, the GS velocity uctuations become large enough andmost of the mean velocity mismatch is removed; better velocity proles are obtained. Simulations for higherReynolds numbers were also carried out with the k model and the one-equation SGS model. Some morework needs to be done to improve the turbulence models and the treatment of the buffer region. However, thenew scheme is effective for reducing the mismatch and is expected to be useful for developing better hybridsimulations.

Appendix

We explain the discretized forms of the Poisson equation and the wall-normal velocity equation as well asthe interpolation of the eddy viscosity in the new scheme. For simplicity, as is the case of Section 3, the de-pendence of physical quantities on the z coordinate is omitted here. Applying the continuity equations (14)and (15) to the NavierStokes equation we have the Poisson equation for the pressure. The Poisson equationcontains the second derivative of the pressure with respect to y; the discretized form at cell (i , j) is given by

2 p5h y2

i, j=

1h2 y p

5hi, j+1 2 p

5hi, j + p5hi, j1 . (A1)

The pressure

p5hi, j

+1 is not originally dened at point ( i, j

+1) ; like the wall-normal velocity it is given by

p5hi, j+1 = ph i, j+1 . (A2)In addition to the streamwise and spanwise velocity components, the transport equation for the wall-normalcomponent v is modied as follows:

vhi, j+1/ 2t =

1h y v

h vh i, j+1 vh vh i, j + , (A3)

v5hi, j1/ 2t =

1h y v

h vh i, j v5h v5h i, j1 + , (A4)


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402 F. Hamba

where

vh vh i, j = vhi, j+1/ 2 +v

5hi, j1/ 2

2

4. (A5)

Not only the discretized forms of the convection term but also those of the SGS stress term are modied. TheSGS stress term involves the eddy viscosity. Although the eddy viscosity is evaluated at the center of a cell,its value at the node is required to calculate the SGS stress term. For example, for node i +12 , j +12 it isusually given by the interpolation

t i+1/ 2, j+1/ 2 = t i+1, j+1 + t i , j+1 + t i+1, j + t i , j

4. (A6)

This value is necessary for the transport equation for u hi+1/ 2, j+1 in the LES region. However, the terms t i+1, j and t i , j in (A6) are evaluated in the RANS region; they are much larger than the other terms inthe LES region. Since this interpolation overestimates the eddy viscosity for the LES near the interface, itis replaced by

t i+1/ 2, j+1/ 2 = t i+1, j+1 + t i , j+12 . (A7)

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A Hybrid RANS LES Simulation of Turbulent Channel Flow