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Compressible laminar flow around a wall-mounted cubic obstacle
Computers & Fluids 36 (2007) 949–960 www.elsevier.com/locate/compfluid
Compressible laminar flow around a wall-mounted cubic obstacle A. van Dijk, H.C. de Lange
*
Department of Mechanical Engineering (wh.2.126), Technische Universiteit Eindhoven, 5600 MB Eindhoven, The Netherlands Received 11 May 2005; received in revised form 18 January 2006; accepted 2 May 2006 Available online 19 January 2007
Abstract The fully-compressible, viscous and non-stationary Navier–Stokes equations are solved for the subsonic flow over a block placed on the floor of a channel. The Reynolds number is varied from 50 to 250. The Mach number is varied between 0.1 and 0.6. In all cases studied the flow field proves to be steady. Several distinct flow features are identified: a horseshoe vortex system, inward bending flow at the side walls of the obstacle, a horizontal vortex at the downstream upper-half of the obstacle and a downstream wake containing two counter-rotating vortices. The shape and size of these flow features are mainly dominated by the Reynolds number. For higher Reynolds numbers, both the horseshoe vortex and the wake region extend over a significantly larger area. The correlation of the position of the separation and attachment point with the Reynolds number has been calculated. Increasing the Mach number (at a fixed Reynolds number of 150) shows its influence in the reduced size, due to compression, of both the wake region and the horseshoe vortex. 2006 Elsevier Ltd. All rights reserved.
1. Introduction Flow separation in internal flows caused by the presence of a wall-mounted obstacle is of particular relevance to numerous practical engineering applications, due to the associated influence of the pressure loss, heat and mass transfer and the resulting effects of erosion and/or corrosion [3]. Experimental and computational research of flow over three-dimensional cubic bodies mainly focuses on high Reynolds numbers, e.g. [5,15,11]. For gaseous fluids these studies are typically applicable for large scale objects in low-speed (incompressible) flow. Less is known for the laminar low-Reynolds-number flow, especially for subsonic situations, where the speeds are high and the objects are small. In this research, the flow around a cubic obstacle in a rectangular channel is studied numerically for Reynolds numbers between 50 and 250. Furthermore, the effect of compressibility is studied for subsonic flow conditions at Mach numbers ranging from Ma = 0.1 to Ma = 0.6. In the low-Mach limit, flows with these characteristics can be found in micro heat exchangers, as they are present in pulse *
Corresponding author. Tel.: +31 2472129; fax: +31 2433445. E-mail address: [email protected] (H.C. de Lange).
0045-7930/$ - see front matter 2006 Elsevier Ltd. All rights reserved. doi:10.1016/j.compfluid.2006.05.003
tubes [21], thermo-acoustic heat pumps [1], etc. Another field in which similar conditions can be found are surface defects or turbulence promoters inside a cooling channel. Furthermore, these low-Reynolds-number flows relate to the flow characteristics of surface-mounted obstacles, e.g. [17], and closed channels with multiple obstacles, e.g. [10], relevant to the cooling flow of electronic components. The problem treated here shows similarities to that of laminar boundary-layer separation past an obstacle. For three-dimensional humps and troughs this problem has been studied theoretically for many decades, e.g. [19]. If one would compare the presently studied flow field to those following from that theory, basically the same features are found. There are two major differences between both geometries. First, we treat a closed channel instead of a boundary-layer flow. Second, the theory treats disturbances with dimensions which must be small compared to the boundary-layer thickness, while in the present study they are of the same magnitude. As will be shown, and might be expected, the main features of the flow remain independent of the Mach number to Ma 0.3. At higher Mach numbers (O(0.5)), the flow can be seen as a model for the (film-)cooling of gasturbine blades. In our experimental studies on film-cooling flow
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(e.g. in [7,8]) and in the numerical research as e.g. in [4] the same (horseshoe and wake) vortical structures have been identified. Although there are a number of additional problems to address in this case (a somewhat higher Reynolds number (500–1000), unsteady interaction of the nozzle/ boundary-layer flow, etc.), comprehension of the main features of the flow presented in this paper may also increase the understanding of the complex flow phenomena in film cooling. Similarly, the description of internal cooling channels in turbine blades can benefit from this study. These cooling channels can be provided with turbulence promoters with dimensions in the order of 10% of the channel diameter. The flow in these channels is subsonic with Mach numbers of 0.5 and slightly higher. The Reynolds numbers (based on the turbulence-promoter height) are usually in the order of 200–1000. In [2] an experimental visualization of the flow around a square cylinder mounted on a flat plate is investigated for a Reynolds number of 1000. Three flow zones have been identified: a steady horseshoe-vortex system upstream of the obstacle, two confined vortices on top of the obstacle and regular vortex shedding downstream of the obstacle. In [22] the laminar horseshoe vortices around a cubic obstacle in a channel have been numerically investigated for Reynolds numbers varying from 5 to 1500. They found that for Re > 20, downstream the obstacle, a three-dimensional wake is formed. As long as a parabolic inlet profile was used, a steady horseshoe-vortex system was observed upstream of the obstacle. In the present study, a numerical code developed at the Technische Universiteit Eindhoven is used. The fullycompressible, viscous and non-stationary Navier–Stokes equations are solved using higher-order finite-difference approximations for collocated variables on a uniform Cartesian grid. In this paper, we will first describe the governing compressible-flow equations and the way in which they are solved numerically. After introducing the geometry and the discretization methods, the results of the calculations are evaluated. In these simulations the flow at a Reynolds number of 150 and a Mach number of 0.1 is used as a reference case. The results for this case are examined in detail and the accuracy of the solution is determined. After that the changes in the flow structure at different Reynolds and Mach numbers are studied. The results show that, at the given Mach and Reynolds numbers, the resulting flow pattern consists of a number of steady vortical structures. Their size and shape mainly depends on the Reynolds number. It is shown that at a Mach number of about 0.3 the compressibility starts to influence the flow. 2. Method
extension of the LODI-method described in [18], where this formulation is used to define boundary conditions for compressible viscous flows. In the non-conservative formulation this leads to the following set of equations: X 1 oq 1 ¼ L2 þ ðL1 þ L5 Þ ð1Þ ot c2 2 i i oui 1 1 osij ðL1 L5 Þi L3i L4i ¼ 2qc q oxj ot X os 1 ¼ ðr q þ DÞ L2i ot qT i
ð2Þ ð3Þ
where q is the density, ui is the velocity in direction i, s is the entropy, c is the local speed of sound, sij is the ij-component of the stress matrix (s), q is the heat-flux vector and D is the viscous dissipation ðs : $vÞ. The characteristic waves L are given by op oui L1i ¼ ðui cÞ qc ð4Þ oxi oxi os L2i ¼ ui ð5Þ oxi oui L3i ¼ uj ð6Þ oxj oui ð7Þ L4i ¼ uk oxk op oui L5i ¼ ðui þ cÞ þ qc ð8Þ oxi oxi Here L1i and L5i are the acoustic waves in direction i, L3i and L4i represent the cross-wind convective fluxes and L2i is the convective transport of entropy. To these conservation equations three equations need to be added c q p ¼ p0 expðss0 Þ=Cv ð9Þ q0 op p ð10Þ c2 ¼ ¼c oq s q p T ¼ ð11Þ qRg where p is the pressure, c is the speed of sound, c is the ratio of specific heats and Cv is the specific heat at constant volume. The index 0 indicates a reference condition and the index s indicates constant entropy. In the following, s0 will be set to zero. The fluid properties correspond with airflow: q0 = 1.2 [kg/m3], T0 = 300 [K], Rg = 287 [J/kg K], c = Cp/Cv = 1.4[], l = 1.458 · 106T1.5/(T + 110.4) [N s/m2], Pr ¼ k ¼ 1. Cp is the specific heat at constant pressure, Rg Cp l is the specific gas constant, l the is dynamic viscosity and k is the heat conductivity.
2.1. Governing equations 2.2. Geometry and boundary conditions The compressible non-stationary Navier–Stokes equations are written as a sum of inviscid characteristics combined with viscous stresses and dissipation. This is an
The chosen geometry (see Fig. 1) consists of a cubic block of size h3, placed X = 3h on the floor of a channel
A. van Dijk, H.C. de Lange / Computers & Fluids 36 (2007) 949–960 0 0 þ fi0 þ x2 fiþ1 ¼ x1 fi1
b1 ðfi fi2 Þ þ a1 ðfi fi1 Þ 2 b2 þ a2 ðfiþ1 fi Þ þ ðfiþ2 fi Þ 2
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ð23Þ
with Fig. 1. Geometry.
with height Ly = 2h. The total channel length Lx = 10h. At the inflow a developed (parabolic) laminar flow profile is prescribed. At the outflow, subsonic non-reflecting boundary conditions are imposed similar to [18]. This means that boundary conditions are imposed separately for the characteristic waves (L1) and stresses (os/on). In spanwise direction, periodic boundary conditions are over the channel width Lz = 8h. Unless stated otherwise, the grid consists of 121 · 49 · 96, uniformly distributed, grid points. Hence, the grid spacing is h/12 · h/24 · h/12. • Inflow s þ Rg lnðq=q0 Þ L v L3 ¼ u L w L4 ¼ u L u ub L5 ¼ L1 þ 2qcðu þ cÞ L L2 ¼ u
ð12Þ ð13Þ ð14Þ ð15Þ
• No-slip wall L5 ¼ L1 or L1 ¼ L5
ð16Þ
u¼v¼w¼0 q¼0
ð17Þ ð18Þ
• Outflow cu p p0 L5 2ðu cÞ L1 ¼ cþu L os12 os13 ¼ ¼0 ox ox oq ¼0 ox
ð19Þ ð20Þ ð21Þ
with L a reference length scale (here L = Lx is chosen) and ub the parabolic inlet profile. The inlet flow is characterized by the Reynolds (Re) and Mach (Ma) numbers: Re ¼
q 0 ub h ub Ma ¼ l0 c0
14 1 a2 ¼ ra1 9 1þr 1 1 1 ða2 a1 Þ b2 ¼ b1 b1 ¼ 18 8 9 1 3 2 x1 ¼ ða2 a1 Þ x2 ¼ x1 3 8 3 a1 ¼
ð22Þ
which are based on the mean inlet velocity ub . 2.3. Discretization For the discretization of all characteristic waves, the fifth-order I5(r = 5/9)-scheme proposed in [13] is used:
and r ¼ 59. This scheme is equivalent to the fifth-order scheme proposed in [23] with its asymmetry parameter a = 2. For the acoustic fluxes L1i and L5i the asymmetry is used as an upwinding scheme along the wave-propagation velocity. This ‘acoustic upwinding’ [13] is similar to the scheme proposed by [20]. For all other (dissipative) fluxes the sixthorder central scheme [14] is used. This scheme also follows from Eq. (23) by using r = 1. For all discretizations at the non-periodic boundaries we use a one-sided third-order implicit scheme: 1 1 1 f10 þ 2f20 ¼ 2f 2 2 f1 þ f3 ð24Þ h 2 2 for the boundary point (i, j = 1) and the fourth-order implicit Pade scheme [14] for the second (next-to-the-wall, i,j = 2) point. The points at the opposite (i, j = N and i,j = N 1) wall are treated similarly. On the adiabatic walls, the sixth-order central scheme is used in both the wall and the next-to-the-wall point to calculate the heat flux (qn = koT/on) and its wall-normal derivative (oqn/on) by assuming symmetry of the T-profile across the wall. For time-marching, the split-time step integration proposed in [12] is used. In this split-time integration the integration of each of the fluxes is bound by its own stability criterion, which decreases the calculation time. Furthermore, it is a low-storage method; the explicit time-integrations (third-order Runge–Kutta (RK3) [9] on the smallest (acoustic) time scale and second-order predictor corrector on the convective and diffusive time-levels) use only one storage variable per time-split level. In all simulations presented a steady solution is found. To find these solutions, each of the time steps for the integration of the different fluxes must be below its respective stability limit 1 m 1 1 1 6 max þ þ ð25Þ i;j;k 3 dx2 Dtd dy 2 dz2 1 juj jvj jwj þ þ 6 max ð26Þ i;j;k Dtc dx dy dz 1 juj þ c jvj þ c jwj þ c þ þ 6 fi; j; k ð27Þ Dta dx dy dz where Dtd, Dtc and Dta represent the diffusive, convective and acoustic time step, respectively. The maxi,j,k indicates
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the maximum over all interior grid points and dx, dy and dz are the local grid spacing in x, y and z direction. As long as these criteria are met, the solutions are numerically stable and evolve towards steady flows. 3. Results The flow structure in the vicinity of the block is first analyzed for a Reynolds number of 150. Next, the changes in the flow for lower and higher Reynolds numbers (between 50 and 250) are analyzed. Note that for the chosen resolution, the maximum mesh-based Reynolds number varies from about 4 to 20. In these calculations, the Mach number is set to Ma = 0.1. To evaluate the effect of compressibility, the Re = 150 simulations are repeated higher Mach numbers (upto Ma = 0.6). The results will be presented as streamline plots. Note that streamlines can end and start at the object both due to the three-dimensional character of the flow and the discretization on which the streamline plots are based. The results will be plotted as a function of x*, y* and z*, which are defined as x/h, y/h and z/h, with x* = 0 at the inflow, y* = 0 at the channel floor and z* = 0 at the object center-line.
The horseshoe vortex that forms around the base of the cube is clearly visible. A small second horseshoe vortex, upstream of the larger one, can also be identified. From these horseshoe vortices, the flow wraps around the cube. Downstream of the cube, two symmetric structures are visible in the wake. Furthermore, vortex structures emerge from the top and side walls of the cube. Fig. 3 shows streamlines for u and w on horizontal crosssections at different vertical positions. The flow field is fully symmetric around the plane z* = 0; therefore, only region z* 6 0 is plotted. The horseshoe vortex upstream of the cube is clearly visible. On the lower cross-sections, the flow reverses directly upstream of the cube whereupon it bends around the obstacle. The two circulation structures downstream of the cube originate from the detached flow at the upstream edges of the cube and not from the horseshoe
3.1. Three-dimensional flow phenomena The occurring three-dimensional flow phenomena are analyzed in detail for Re = 150 and Ma = 0.1. To give a first impression of the flow, the three-dimensional iso-surfaces for k2 are presented in Fig. 2. Here, k2 is the second eigenvalue of X2 + S2, where X and S are the anti-symmetric and symmetric part of the velocity-gradient matrix [6].
Fig. 2. Top (top) and head (bottom) view of k2 iso-surface for Re = 150.
Fig. 3. Streamlines for Re = 150 and Ma = 0.1 at y* = 0.04, 1/4, 1/2 and 11/12 (from top to bottom).
A. van Dijk, H.C. de Lange / Computers & Fluids 36 (2007) 949–960
vortex. The horseshoe flow bends around the circulation areas and slightly expands again in the downstream wake behind the cube. It can be seen that the horseshoe vortex becomes thinner for larger y*. The horseshoe vortex has vanished at y* = 1/2, while the circulation zones behind the obstacle are still present. The structures emerging from the side walls of the obstacle and bending around it also remain present. At y ¼ 11 , the reversed flow at the back of the obstacle is still 12 visible. The side-wall structures no longer are closed vortices at this height. In Fig. 4, streamlines for u and v on four vertical crosssections (different z*-values) are displayed. Two prominent features can be recognized. First, the horseshoe vortex at the upstream corner of the channel floor and the obstacle is again clearly visible. Second, these cross-sections show the circulation zone downstream of the object. On the cross-section that coincides with the plane of symmetry (z* = 0), the center of the circulation zone is located behind the obstacle at approximately x* = 5 and y* = 0.8. The reversed flow is directed upwards near the top of the object, while a downward motion is found at the base of the obstacle. The attachment line is also visible, starting at x* ’ 6.4 for y* = 0. At z* = 1/4, the circulation zone no longer is a closed structure. At z* = 5/12, the circulation zone has disappeared, only the horseshoe vortex remains present. For 0.4 < y* < 1, the downstream flow moves upwards, past
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the circulation zone that is present near the obstacle center line. At z* = 7/12, directly next to the obstacle, the horseshoe vortex in front of the obstacle is shows as a slight upward motion directly above the channel floor. 3.2. Numerical accuracy To estimate the accuracy of the presented results, the calculation of the Re = 150 and Ma = 0.1 flow (as presented in the previous paragraph) has been repeated with double resolution in x, y or z direction. In Fig. 5 the streamlines at y* = 0.04 (near to the wall) resulting from two of these simulations can be found. In this figure the solution on the x-wise refined grid with 241 · 49 · 96 nodes (grid spacing: h/24 · h/24 · h/12) and z-wise refined with 121 · 49 · 192 nodes (grid spacing: h/12 · h/24 · h/24) are shown. These results show that the numerical resolution on the 121 · 49 · 96 grid is sufficient to capture all relevant flow features. To quantify the influence of grid refinement, the solution of the base (121 · 49 · 96) grid is compared to a number of more refined grids in Table 1. The first three columns in this table give the root-mean-square (rms) differences of u, v and w sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 RN s ðfr fs Þ2 rf ¼ ð28Þ U1 Ns
Fig. 4. Streamlines for Re = 150 and Ma = 0.1 at z* = 0, 1/4, 5/12 and 7/12 (from top to bottom).
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Fig. 5. Streamlines for Re = 150 and Ma = 0.1 at y* = 0.04 for the refined x-grid (top) and z-grid (bottom).
Table 1 The rms (ru, rv and rw defined in Eq. (28)) and the distance of the separation and attachment point to the obstacle (respectively xS and xR) for the different resolutions Grid
ru [103]
rv [103]
rw [103]
xS
xR
61 · 25 · 48 •121 · 49 · 96 241 · 49 · 96 121 · 97 · 96 121 · 49 · 192 241 · 49 · 192 241 · 97 · 192
39 10 21 9 9 5 –
9 3 4 3 2 1 –
10 3 5 3 4 1 –
0.95 1.04 1.06 1.04 1.02 1.023 1.021
2.79 2.46 2.57 2.44 2.37 2.407 2.405
The symbol • indicates the base grid, i.e. the grid used in the remainder of the paper.
where f is either u, v or w, the subscript r indicates the finest-grid (241 · 97 · 192) solution, while the subscript s indicates the solution on the studied grid. Ns is the total number of grid-points on the studied (s)-grid. Clearly, the rms-differences between the base-grid (121 · 49 · 96) solution and that of the more refined grid is small (one percent and less). The largest effect of refinement is on the main-stream velocity (u), while both other velocity components are predicted more accurately. Table 1 also shows that grid-refinement in x-direction only appears to decrease the accuracy of the solution. Furthermore, from the rms-values it follows that refinement in y- and z-direction has less influence. However, when refinement in x- and z-direction are performed simultaneously the solution closely resembles that of the finest grid. To show global grid convergence of the studied solutions the solution on a coarser grid (61 · 25 · 48) has been added. The rms-differences of this solution are in the order of 4% for u and 1% for v and w. Comparing this to the rmsdifferences of the base grid shows that the global conver-
gence of the solution is approximately quadratic, which means that the convergence is determined by the accuracy of the wall-point discretization. For the studied flow this is of course not a surprise, since all the flow structures, as described in the previous section, are in some way wallbounded. In the remainder of this paper, the separation point in front of the obstacle and the attachment point in its wake are being studied. These points will be defined as the far upstream point and the far downstream point on the centerline (y* = 0 and z* = 0) where (ou*/oy*) = 0. The distances between these points and the facing side of the block (xS for the upstream point and xR for the downstream point) are presented in Table 1 for all grids considered. This shows that the spacing extent of the horseshoe vortex (within 2%) and the wake (within 5%) barely depends on the chosen grid. Again, x- and z-refinement show the largest effect. It is worthwhile to mention that the effect of x- and z-refinement on xS and xR is opposite. For the x-wise refined grid both xS and xR increase, while for the z-wise refined grid they both decrease. When x-wise and z-wise refinement are performed simultaneously, the effects more or less cancel. Therefore, the accuracy of the predicted xS and xR is indeed in the order of 1–2%. The effect of y-wise refinement is much less. This also shows on the more refined level; when comparing the 241 · 49 · 192 and 241 · 97 · 192 grids only marginal differences are seen. To appreciate the accuracy of the solution it is worth remembering that the grid spacing on the base grid equals h/12 · h/24 · h/12. This means that the differences between xR and xS on the different grids (except for the coarsest grid) are well within the grid spacing. In the remainder of this paper, the chosen base grid (121 · 49 · 96 nodes, grid spacing: h/12 · h/24 · h/12) is assumed to provide sufficient resolution for accurately representing the flow. It is worthwhile noting that the changes in xS and xR due to the effect of Reynolds and Mach number (as they will be presented later in this paper) are considerably larger. 3.3. Effect of Reynolds number Fig. 6 shows the results for the u-w streamlines directly above the channel floor (y* = 0.04) Reynolds numbers ranging from Re = 50 to Re = 250. It can be seen that for higher Reynolds numbers, both upstream and downstream vortex structures increase in size. For Re = 50, the downstream circulation emerges from the upstream vortex. For Re = 100, it emerges from both the upstream vortex and the flow that is detached at the upstream obstacle edges. For Re = 150 and higher, the downstream circulation emerges from the detached flow at the obstacle edges. The horseshoe flow bends around these structures. For Re = 250, the second horseshoe vortex structure upstream of the first horseshoe is clearly visible. This second structure bends around the first horseshoe vortex and merges into it in the wake region.
A. van Dijk, H.C. de Lange / Computers & Fluids 36 (2007) 949–960
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upstream of the object decays Re = 250. For Re = 50, ou oy monotonically from its value at the inlet up to a local minimum value at X* 2. For Re = 250, the derivative has three roots (ou*/oy* = 0) before it vanishes at the obstacle. The first dip corresponds with the small upstream vortex. Next, there is a small positive gradient where there exists a small secondary vortex in between of both primary horseshoe structures. The large dip in the velocity gradient close to the object corresponds with the major large horseshoe vortex. Downstream (x > 4) of the obstacle ou exhibits a local oy maximum followed by a local minimum in the case of Re = 250. The local maximum is present for the higher Reynolds numbers. It reflects the reversed flow structure at the downstream lower corner of the obstacle. For Re = 50 this local maximum is absent, implying that the reversed flow structure is not present. In Fig. 9 the correlation of xS and xR (as defined in Section 3.2) with the Reynolds number is presented. Both figures suggest a logarithmic correlation between the Reynolds number to the length of both zones. In our case (a closed channel) xR Re0.85. In [17] the length of the circulation flow of an obstacle on a flat plate in a boundary layer flow (with a boundary-layer thickness d equal to h) was studied numerically. They concluded that, for conditions similar to those presented here xR Re0.684. The slight difference between the study in [17] and our results may be due to the slightly different geometries (a closedchannel geometry in our study compared to the boundary layer in [17]). 3.5. Compressibility effect
Fig. 6. Streamlines (Ma = 0.1) at y* = 0.04 for Re = 50, 100, 200 and 250 (from top to bottom).
Fig. 7 shows, like Fig. 6, that the upstream and downstream vortices grow in size with increasing Reynolds number. The second horseshoe vortex, situated upstream of the first one, is clearly visible for Re = 250. The vortex structure downstream of the upper half of the obstacle is no longer closed for Re = 250. For Re = 50 and Re = 100 the downward flow, as found for Re = 150 and higher, is no longer present at the downstream junction of the channel floor and the obstacle. 3.4. Separation and attachment points The growing number of structures for increasing Reynolds number is also reflected by the ou*/oy*-derivative (with u u=ub ) on the channel floor at the z-plane center line. In Fig. 8 this derivative is given upstream (x < 3) and downstream (x > 4) of the object for Re = 50 and
To judge whether the flow depends on the Mach number for subsonic conditions, the effect of compressibility is evaluated by comparing the results at Re = 150 and Ma = 0.1 with those for Ma = 0.5 and Ma = 0.6. The results of these simulations are presented in Figs. 10 and 11. From the streamlines close to the channel floor, as presented in Fig. 10, it is clear that the results for Ma = 0.1 and 0.3 are very similar. There appears to be a small change of the flow pattern just ahead of the block. The vortical structure shows a slightly stronger downward motion close to the leading edge. Moreover, the secondary motion at the top of the horseshoe vortex seems to increase with the Mach number. These observations lead to the conclusion that the horseshoe vortex may become somewhat stronger with increasing Mach number. When the results for Ma = 0.6 are taken into consideration, the growing strength of the vortex with increasing Mach numbers becomes more obvious. The reason for this growth may lie in the temperature in the stagnation point. As the Mach number increases, so does the stagnation temperature. Now, if we would assume that the compression ahead of the block is approximately isentropic, we can show that the kinematic viscosity (m = l/q) in the horseshoe vortex must decrease with increasing Mach number. This leads
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Fig. 7. Streamlines (Ma = 0.1) at z* = 0 for Re = 50, 100, 200 and 250 (from top to bottom).
to reduced diffusion, which induces an increased vortex strength. Another striking change can be seen in the downstream region (in Fig. 10). The higher Mach number flow clearly exhibits a reduced distance xR to the downstream attachment point. Clearly, the circulation area behind the block becomes more compact, indicating an increased lateral flow. The flow is obviously sucked in from the sides due to the increased dynamic pressure. Both flow features are also shown in Fig. 11. This figure gives the streamlines on the centerplane z* = 0. Again, for Ma = 0.3 and 0.6 the backflow is found to be somewhat larger indicating an increased strength of the vortex. The contours also show that the spatial extent of the influence of the block expands further in upstream direction. On the other hand, in the downstream region the wake narrows. When the influence of the Mach number on xS and xR is studied, we find the results as depicted in Fig. 12. It is clear that the effect of the Mach number on both the separation and attachment point is much less than that of the Reynolds number. For the considered Mach numbers, the (downward) compression of the horseshoe vortex leads to a slight (20%) increase of the length of the separation zone from xS 1.04 at Ma = 0.1 to xS 1.23 at Ma = 0.6. As is apparent from Fig. 11 the attachment point moves closer to
the block with increasing Mach number. This leads to a 10% (linear) decrease of xR from xR 2.46 at Ma = 0.1 to xR 2.26 at Ma = 0.6. It is worthwhile noting that, although the Mach number has only a marginal effect on xS and xR, the changes of 10–20% are still well beyond the numerical accuracy (1–2%). Fig. 13 gives the streamwise velocity on the obstacle centerline at y* = 0.04 in front (top) and behind (bottom) the obstacle for increasing Mach numbers (0.1, 0.3 and 0.6). As can be seen from Fig. 13 (top) the increase of xS with higher Mach numbers goes hand in hand with the appearance of the ‘double’ vortex structure; just as in the case of the increasing Reynolds number. The profile behind the object (Fig. 13, bottom) shows that the velocity increases much faster for higher Mach numbers. This, again, indicates that due to the increased dynamical pressure there is a larger lateral inflow of fluid, thus leading to a faster acceleration. 3.6. Validation of the results In [2] the flow around a square cylinder mounted on a flat plate has been studied experimentally for Re = 1000 (based on a uniform inflow velocity and the cylinder diameter). Smoke visualizations and Particle Image Velocimetry measurements are adopted. The main differences between their set-up and the numerical set-up presented in this
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1.4
1
0
xS
(∂u*/∂y*)y*=0
1.2
1
–1 0.8
0
2
4
6
8
10
x*
102
Reh
102
Re
3
2.5
–5
xR
(∂u*/∂y*)y*=0
0
2
–10
0
2
4
6
8
10
x*
1.5 h
Fig. 8. ðou =oy Þy ¼z ¼0 for Re = 50 (top) and Re = 250 (bottom) (Ma = 0.1).
Fig. 9. Correlation of xS (top) and xR (bottom) with the logarithm of the Reynolds number (Ma = 0.1).
paper is their uniform inflow velocity profile and the absence of a top channel wall. Furthermore, the square cylinder height equals 0.3 times its diameter, in contrast to the cubic geometry in our study. For higher Reynolds numbers, the flow field around several prismatic obstacles has been experimentally investigated in e.g. [16]. These experiments are performed in fully developed channel flow. The Reynolds number, based on the height of the channel and the mean bulk velocity, lies between 8 · 104 and 1.2 · 105. Several distinct similarities can be observed between these experimental results and our numerical results. Similar to the structures visible in Fig. 2, the visualization studies show one large and one smaller horseshoe vortex structure. Furthermore, for both [2] and our numerical results, a small reversed flow structure (leading to the bump in ou/oy in Fig. 8 for Re = 250) is present at the channel floor. In [22] the laminar flow between two parallel plates with a cube mounted on the lower wall is numerically investigated. The Reynolds number, based on the bulk mean velocity and the obstacle height, is varied from Re = 5 to Re = 1500. The length of each side of the cubic obstacle
is fixed as one half of the channel height (h/H = 1/2). The streamwise and spanwise domain lengths equal 10h and 7h, respectively. For Reynolds numbers Re > 250, the streamwise domain length is increased by 5h. Periodic boundary conditions are applied in spanwise direction. At all solid walls, no-slip boundary conditions is imposed. A parabolic velocity profile is prescribed at the inlet. In [22] a clear drawing of the u and w streamlines directly above the channel floor is presented for Re = 350. It shows a steady symmetric flow profile around the z-plane centerline. The inward-bending flow at the cube edges and the horseshoe vortex upstream of the cubic obstacle are also present similar to those visible in Fig. 6. A striking difference between the results in [22] and those of this study is the topology of the wake. In [22] the two downstream circulation structures cannot be identified. It must be noted that our results are more in line with the experimental results by [2] where the two contra-rotating vortices with vertical axis remain confined behind the obstacle have also been observed. Furthermore, our wake results bear a striking resemblance with the close-to-the-wall results presented in [4]. In the latter numerical study large eddy simulations are performed on the film-cooling problem. Although the
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parameters to that problem are somewhat different to ours, the wake region in [4] also shows the closed double vortex system, which characterizes the wake as we find it. In [22] the relation between the Reynolds number and the distance between the far upstream separation point and the upstream face of the obstacle (xS) is presented. Their results are compared with the present simulations in Fig. 14. A notable feature in this figure is the jump found in the results of [22] at Re = 350. In [22] this is attributed to the changing nature of the flow at Re = 350. However, the jump also coincides with an elongation of their computational domain by a factor 1.5. Our results seem to correspond very well to those of the elongated domain in [22]. This might suggest that for the shorter domain the outflow boundary conditions used in [22] still have some influence on the attachment point. 4. Conclusions
Fig. 10. Streamlines (Re = 150) at y* = 0.04 for Ma = 0.1, 0.3 and 0.6 (from top to bottom).
The subsonic laminar flow around a cubic obstacle placed on the floor of a spanwise periodic channel has been studied numerically. As a first step a base solution is described, it consists of a flow at Re = 150 and Ma = 0.1 solved on a grid of 121 · 49 · 96 nodes. Four main flow zones are identified: (i) a horseshoe-vortex system, (ii) inward bending flow at the side walls of the obstacle, (iii) a vortex with a horizontal axis at the downstream upper half of the obstacle and (iv) a downstream wake containing two counter-rotating vortices with vertical axes. When the solution of the base grid (121 · 49 · 96 nodes) is compared to a refined grid (241 · 97 · 192 nodes) the length of the
Fig. 11. Streamlines (Re = 150) at z* = 0 for Ma = 0.1, 0.3 and 0.6 (from top to bottom).
A. van Dijk, H.C. de Lange / Computers & Fluids 36 (2007) 949–960 1.25
959
0.2 0.1
1.2
0 –0.1
1.15
x
S
u*
–0.2 –0.3
1.1
–0.4 –0.5 1.05
1 0.1
0.15
0.2
0.25
0.3
0.35
0.4
0.45
0.5
0.55
0.6
–0.6
Ma=0.1 Ma=0.3 Ma=0.6
–0.7 0
0.5
1
1.5
2
2.5
3
8
9
10
x*
Ma 0.25
2.5
0.2
Ma=0.1 Ma=0.3 Ma=0.6
2.45
0.15 2.4
x
u*
R
0.1 0.05
2.35
0 2.3
2.25 0.1
–0.05 –0.1 4 0.15
0.2
0.25
0.3
0.35
0.4
0.45
0.5
0.55
5
6
7
x*
0.6
Ma
1.8
1.4 S
separation and circulation zones are predicted with an accuracy of 2% and 5%, respectively. Starting from the base solution, the effect of the Reynolds number and the Mach number is studied. The Reynolds number is increased in four steps from Re = 50 to Re = 250 with the Mach number fixed at Ma = 0.1. Next, the Mach number is increased to Ma = 0.3 and Ma = 0.6 with a fixed Reynolds number at Re = 150. For all cases investigated, a steady flow field is found. For Reynolds numbers Re P 150, three additional flow structures are formed: (i) an additional horseshoe vortex, (ii) a vortex with a horizontal axis on top of the obstacle and (iii) a vortex at the downstream junction of the obstacle and the channel floor. The latter structure counterrotates the vortex at the downstream upper half of the obstacle. For Re = 250, the vortex at the downstream upper half of the obstacle no longer forms a closed structure. A logarithmic correlation is found between the Reynolds number and the distance between the block and the far upstream separation point (xS) and the far downstream attachment point (xR).
Fig. 13. u* above the plate at the centerline (Re = 150) for Ma = 0.1, 0.3 and 0.6 in front (top) and behind (bottom) the object.
x
Fig. 12. Correlation of xS (top) and xR (bottom) with the Mach number (Re = 150).
1
0.6
1
10
2
3
10
10
Re
h
Fig. 14.umThe correlation of xS with the logarithm of the Reynolds number Reh ¼ m h , comparing the results in [22] (o) with the present results (+).
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When increasing the Mach number it becomes obvious that (at these subsonic conditions) the flow pattern is highly Re-dominated. There are however, some differences detected. The major points are an increase in the strength of the horseshoe vortex, an increased upstream influence of the obstruction and a decreased length of the circulation zone. It appears that the latter two are caused by the increase of the pressure differences due to the increased dynamic pressure. The slight increase in the strength of the horseshoe vortex is attributed to the local decrease in kinematic viscosity. References [1] Bosma AJ, Stoffels GGM, van der Meer TH. Numerical simulation of heat transfer of nickel foam in thermo-acoustic waves. In: Proceedings of the EuroTherm74: Heat transfer in unsteady and transitional flows, 2003. pp. 79–84. [2] Calluaud D, David L, Texier A. Study of the laminar flow around a square cylinder, Ninth international symposium on flow visualization, 2000; paper 402. pp. 6. [3] Dimaczek G, Kessler R, Martinuzzi R, Tropea C. The flow over twodimensional, surface mounted obstacles at high Reynolds numbers. In: Seventh symposium on turbulent shear flows. vol. 10(1), 1989. pp. 1–6. [4] Guo X, Schro¨der W, Meinke M. Large-eddy simulations of filmcooling flows. Comput Fluids 2006;35:587–606. [5] Hussein HJ, Martinuzzi RJ. Energy balance for turbulent flow around a surface mounted cube placed in a channel. Phys Fluids 1996;8:764–80. [6] Jeong J, Hussain F. On the identification of a vortex. J Fluid Mech 1995;285:69–94. [7] Jovanovic MB, de Lange HC, van Steenhoven AA. Influence of laser drilling imperfection on film cooling performances, ASME paper, 2005; GT2005-68251. [8] Jovanovic MB, de Lange HC, van Steenhoven AA. Influence of hole imperfection on jet-cross flow interaction. Int J Heat Fluid Flow 2006;27:42–53.
[9] Kennedy CA, Carpenter MH, Lewis RM. Low-storage, explicit Runge–Kutta schemes for compressible Navier–Stokes equations. Appl Num Math 2000;35:177–219. [10] Korichi A, Oufer L. Numerical heat transfer in a rectangular channel with mounted obstacles on upper and lower walls. Int J Therm Sci 2005;44:644–55. [11] Krajnovic S, Davidson LV. Flow around a three-dimensional bluff body. In: Ninth international symposium on flow visualization, 2000; paper 177. pp. 10. [12] de Lange HC. Split time-integration for Low-Mach Number compressible flows. Comm Numer Meth Eng 2004;20(7):501–9. [13] de Lange HC. Inviscid flow modelling using asymmetric implicit finite difference schemes. Int J Numer Meth Fluids 2005;49: 1033–51. [14] Lele SK. Compact finite difference schemes with spectral-like resolution. J Comput Phys 1992;103:16–42. [15] Martinuzzi R, Melling A, Tropea C. Reynolds stress field for the turbulent flow around a surface-mounted cube placed in a channel. In: Ninth symposium on turbulent shear flows, 1993. 13-4-1. [16] Martinuzzi R, Tropea C. The flow around surface-mounted, prismatic obstacles placed in fully developed channel flow. J Fluids Eng 1993;115:85–91. [17] Ngo Boum GB, Martemianov S, Alemany A. Computational study of laminar flow and mass transfer around a surface-mounted obstacle. Int J Heat Mass Transfer 1999;42:2849–61. [18] Poinsot TJ, Lele SK. Boundary conditions for direct simulations of compressible viscous flows. J Comput Phys 1992;101:104–29. [19] Roget A, Brazier JP, Cousteix J, Mauss J. A contribution to the physical analysis of separated flows past three-dimensional humps. Eur J Mech B/Fluids 1998;17:307–29. [20] Sesterhenn J. A characteristic-type formulation of the Navier–Stokes equations for high order upwind schemes. Comput Fluids 2001;30: 37–67. [21] Tanaeva I. Low-temperature cryocooling. PhD-thesis, Technische Universiteit Eindhoven, 2004. [22] Yang KS, Hwang JY. Numerical study of horseshoe vortices around a cubic obstacle in a channel, ASME 2000 Fluids Engineering Division Summer Meeting, 2000; FEDSM00-11236. [23] Xiaolin Zhong. High-order finite-difference schemes for numerical simulation of hypersonic boundary layer transition. J Comput Phys 1998;144:662–709.
Compressible laminar flow around a wall-mounted cubic obstacle A. van Dijk, H.C. de Lange
*
Department of Mechanical Engineering (wh.2.126), Technische Universiteit Eindhoven, 5600 MB Eindhoven, The Netherlands Received 11 May 2005; received in revised form 18 January 2006; accepted 2 May 2006 Available online 19 January 2007
Abstract The fully-compressible, viscous and non-stationary Navier–Stokes equations are solved for the subsonic flow over a block placed on the floor of a channel. The Reynolds number is varied from 50 to 250. The Mach number is varied between 0.1 and 0.6. In all cases studied the flow field proves to be steady. Several distinct flow features are identified: a horseshoe vortex system, inward bending flow at the side walls of the obstacle, a horizontal vortex at the downstream upper-half of the obstacle and a downstream wake containing two counter-rotating vortices. The shape and size of these flow features are mainly dominated by the Reynolds number. For higher Reynolds numbers, both the horseshoe vortex and the wake region extend over a significantly larger area. The correlation of the position of the separation and attachment point with the Reynolds number has been calculated. Increasing the Mach number (at a fixed Reynolds number of 150) shows its influence in the reduced size, due to compression, of both the wake region and the horseshoe vortex. 2006 Elsevier Ltd. All rights reserved.
1. Introduction Flow separation in internal flows caused by the presence of a wall-mounted obstacle is of particular relevance to numerous practical engineering applications, due to the associated influence of the pressure loss, heat and mass transfer and the resulting effects of erosion and/or corrosion [3]. Experimental and computational research of flow over three-dimensional cubic bodies mainly focuses on high Reynolds numbers, e.g. [5,15,11]. For gaseous fluids these studies are typically applicable for large scale objects in low-speed (incompressible) flow. Less is known for the laminar low-Reynolds-number flow, especially for subsonic situations, where the speeds are high and the objects are small. In this research, the flow around a cubic obstacle in a rectangular channel is studied numerically for Reynolds numbers between 50 and 250. Furthermore, the effect of compressibility is studied for subsonic flow conditions at Mach numbers ranging from Ma = 0.1 to Ma = 0.6. In the low-Mach limit, flows with these characteristics can be found in micro heat exchangers, as they are present in pulse *
Corresponding author. Tel.: +31 2472129; fax: +31 2433445. E-mail address: [email protected] (H.C. de Lange).
0045-7930/$ - see front matter 2006 Elsevier Ltd. All rights reserved. doi:10.1016/j.compfluid.2006.05.003
tubes [21], thermo-acoustic heat pumps [1], etc. Another field in which similar conditions can be found are surface defects or turbulence promoters inside a cooling channel. Furthermore, these low-Reynolds-number flows relate to the flow characteristics of surface-mounted obstacles, e.g. [17], and closed channels with multiple obstacles, e.g. [10], relevant to the cooling flow of electronic components. The problem treated here shows similarities to that of laminar boundary-layer separation past an obstacle. For three-dimensional humps and troughs this problem has been studied theoretically for many decades, e.g. [19]. If one would compare the presently studied flow field to those following from that theory, basically the same features are found. There are two major differences between both geometries. First, we treat a closed channel instead of a boundary-layer flow. Second, the theory treats disturbances with dimensions which must be small compared to the boundary-layer thickness, while in the present study they are of the same magnitude. As will be shown, and might be expected, the main features of the flow remain independent of the Mach number to Ma 0.3. At higher Mach numbers (O(0.5)), the flow can be seen as a model for the (film-)cooling of gasturbine blades. In our experimental studies on film-cooling flow
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(e.g. in [7,8]) and in the numerical research as e.g. in [4] the same (horseshoe and wake) vortical structures have been identified. Although there are a number of additional problems to address in this case (a somewhat higher Reynolds number (500–1000), unsteady interaction of the nozzle/ boundary-layer flow, etc.), comprehension of the main features of the flow presented in this paper may also increase the understanding of the complex flow phenomena in film cooling. Similarly, the description of internal cooling channels in turbine blades can benefit from this study. These cooling channels can be provided with turbulence promoters with dimensions in the order of 10% of the channel diameter. The flow in these channels is subsonic with Mach numbers of 0.5 and slightly higher. The Reynolds numbers (based on the turbulence-promoter height) are usually in the order of 200–1000. In [2] an experimental visualization of the flow around a square cylinder mounted on a flat plate is investigated for a Reynolds number of 1000. Three flow zones have been identified: a steady horseshoe-vortex system upstream of the obstacle, two confined vortices on top of the obstacle and regular vortex shedding downstream of the obstacle. In [22] the laminar horseshoe vortices around a cubic obstacle in a channel have been numerically investigated for Reynolds numbers varying from 5 to 1500. They found that for Re > 20, downstream the obstacle, a three-dimensional wake is formed. As long as a parabolic inlet profile was used, a steady horseshoe-vortex system was observed upstream of the obstacle. In the present study, a numerical code developed at the Technische Universiteit Eindhoven is used. The fullycompressible, viscous and non-stationary Navier–Stokes equations are solved using higher-order finite-difference approximations for collocated variables on a uniform Cartesian grid. In this paper, we will first describe the governing compressible-flow equations and the way in which they are solved numerically. After introducing the geometry and the discretization methods, the results of the calculations are evaluated. In these simulations the flow at a Reynolds number of 150 and a Mach number of 0.1 is used as a reference case. The results for this case are examined in detail and the accuracy of the solution is determined. After that the changes in the flow structure at different Reynolds and Mach numbers are studied. The results show that, at the given Mach and Reynolds numbers, the resulting flow pattern consists of a number of steady vortical structures. Their size and shape mainly depends on the Reynolds number. It is shown that at a Mach number of about 0.3 the compressibility starts to influence the flow. 2. Method
extension of the LODI-method described in [18], where this formulation is used to define boundary conditions for compressible viscous flows. In the non-conservative formulation this leads to the following set of equations: X 1 oq 1 ¼ L2 þ ðL1 þ L5 Þ ð1Þ ot c2 2 i i oui 1 1 osij ðL1 L5 Þi L3i L4i ¼ 2qc q oxj ot X os 1 ¼ ðr q þ DÞ L2i ot qT i
ð2Þ ð3Þ
where q is the density, ui is the velocity in direction i, s is the entropy, c is the local speed of sound, sij is the ij-component of the stress matrix (s), q is the heat-flux vector and D is the viscous dissipation ðs : $vÞ. The characteristic waves L are given by op oui L1i ¼ ðui cÞ qc ð4Þ oxi oxi os L2i ¼ ui ð5Þ oxi oui L3i ¼ uj ð6Þ oxj oui ð7Þ L4i ¼ uk oxk op oui L5i ¼ ðui þ cÞ þ qc ð8Þ oxi oxi Here L1i and L5i are the acoustic waves in direction i, L3i and L4i represent the cross-wind convective fluxes and L2i is the convective transport of entropy. To these conservation equations three equations need to be added c q p ¼ p0 expðss0 Þ=Cv ð9Þ q0 op p ð10Þ c2 ¼ ¼c oq s q p T ¼ ð11Þ qRg where p is the pressure, c is the speed of sound, c is the ratio of specific heats and Cv is the specific heat at constant volume. The index 0 indicates a reference condition and the index s indicates constant entropy. In the following, s0 will be set to zero. The fluid properties correspond with airflow: q0 = 1.2 [kg/m3], T0 = 300 [K], Rg = 287 [J/kg K], c = Cp/Cv = 1.4[], l = 1.458 · 106T1.5/(T + 110.4) [N s/m2], Pr ¼ k ¼ 1. Cp is the specific heat at constant pressure, Rg Cp l is the specific gas constant, l the is dynamic viscosity and k is the heat conductivity.
2.1. Governing equations 2.2. Geometry and boundary conditions The compressible non-stationary Navier–Stokes equations are written as a sum of inviscid characteristics combined with viscous stresses and dissipation. This is an
The chosen geometry (see Fig. 1) consists of a cubic block of size h3, placed X = 3h on the floor of a channel
A. van Dijk, H.C. de Lange / Computers & Fluids 36 (2007) 949–960 0 0 þ fi0 þ x2 fiþ1 ¼ x1 fi1
b1 ðfi fi2 Þ þ a1 ðfi fi1 Þ 2 b2 þ a2 ðfiþ1 fi Þ þ ðfiþ2 fi Þ 2
951
ð23Þ
with Fig. 1. Geometry.
with height Ly = 2h. The total channel length Lx = 10h. At the inflow a developed (parabolic) laminar flow profile is prescribed. At the outflow, subsonic non-reflecting boundary conditions are imposed similar to [18]. This means that boundary conditions are imposed separately for the characteristic waves (L1) and stresses (os/on). In spanwise direction, periodic boundary conditions are over the channel width Lz = 8h. Unless stated otherwise, the grid consists of 121 · 49 · 96, uniformly distributed, grid points. Hence, the grid spacing is h/12 · h/24 · h/12. • Inflow s þ Rg lnðq=q0 Þ L v L3 ¼ u L w L4 ¼ u L u ub L5 ¼ L1 þ 2qcðu þ cÞ L L2 ¼ u
ð12Þ ð13Þ ð14Þ ð15Þ
• No-slip wall L5 ¼ L1 or L1 ¼ L5
ð16Þ
u¼v¼w¼0 q¼0
ð17Þ ð18Þ
• Outflow cu p p0 L5 2ðu cÞ L1 ¼ cþu L os12 os13 ¼ ¼0 ox ox oq ¼0 ox
ð19Þ ð20Þ ð21Þ
with L a reference length scale (here L = Lx is chosen) and ub the parabolic inlet profile. The inlet flow is characterized by the Reynolds (Re) and Mach (Ma) numbers: Re ¼
q 0 ub h ub Ma ¼ l0 c0
14 1 a2 ¼ ra1 9 1þr 1 1 1 ða2 a1 Þ b2 ¼ b1 b1 ¼ 18 8 9 1 3 2 x1 ¼ ða2 a1 Þ x2 ¼ x1 3 8 3 a1 ¼
ð22Þ
which are based on the mean inlet velocity ub . 2.3. Discretization For the discretization of all characteristic waves, the fifth-order I5(r = 5/9)-scheme proposed in [13] is used:
and r ¼ 59. This scheme is equivalent to the fifth-order scheme proposed in [23] with its asymmetry parameter a = 2. For the acoustic fluxes L1i and L5i the asymmetry is used as an upwinding scheme along the wave-propagation velocity. This ‘acoustic upwinding’ [13] is similar to the scheme proposed by [20]. For all other (dissipative) fluxes the sixthorder central scheme [14] is used. This scheme also follows from Eq. (23) by using r = 1. For all discretizations at the non-periodic boundaries we use a one-sided third-order implicit scheme: 1 1 1 f10 þ 2f20 ¼ 2f 2 2 f1 þ f3 ð24Þ h 2 2 for the boundary point (i, j = 1) and the fourth-order implicit Pade scheme [14] for the second (next-to-the-wall, i,j = 2) point. The points at the opposite (i, j = N and i,j = N 1) wall are treated similarly. On the adiabatic walls, the sixth-order central scheme is used in both the wall and the next-to-the-wall point to calculate the heat flux (qn = koT/on) and its wall-normal derivative (oqn/on) by assuming symmetry of the T-profile across the wall. For time-marching, the split-time step integration proposed in [12] is used. In this split-time integration the integration of each of the fluxes is bound by its own stability criterion, which decreases the calculation time. Furthermore, it is a low-storage method; the explicit time-integrations (third-order Runge–Kutta (RK3) [9] on the smallest (acoustic) time scale and second-order predictor corrector on the convective and diffusive time-levels) use only one storage variable per time-split level. In all simulations presented a steady solution is found. To find these solutions, each of the time steps for the integration of the different fluxes must be below its respective stability limit 1 m 1 1 1 6 max þ þ ð25Þ i;j;k 3 dx2 Dtd dy 2 dz2 1 juj jvj jwj þ þ 6 max ð26Þ i;j;k Dtc dx dy dz 1 juj þ c jvj þ c jwj þ c þ þ 6 fi; j; k ð27Þ Dta dx dy dz where Dtd, Dtc and Dta represent the diffusive, convective and acoustic time step, respectively. The maxi,j,k indicates
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the maximum over all interior grid points and dx, dy and dz are the local grid spacing in x, y and z direction. As long as these criteria are met, the solutions are numerically stable and evolve towards steady flows. 3. Results The flow structure in the vicinity of the block is first analyzed for a Reynolds number of 150. Next, the changes in the flow for lower and higher Reynolds numbers (between 50 and 250) are analyzed. Note that for the chosen resolution, the maximum mesh-based Reynolds number varies from about 4 to 20. In these calculations, the Mach number is set to Ma = 0.1. To evaluate the effect of compressibility, the Re = 150 simulations are repeated higher Mach numbers (upto Ma = 0.6). The results will be presented as streamline plots. Note that streamlines can end and start at the object both due to the three-dimensional character of the flow and the discretization on which the streamline plots are based. The results will be plotted as a function of x*, y* and z*, which are defined as x/h, y/h and z/h, with x* = 0 at the inflow, y* = 0 at the channel floor and z* = 0 at the object center-line.
The horseshoe vortex that forms around the base of the cube is clearly visible. A small second horseshoe vortex, upstream of the larger one, can also be identified. From these horseshoe vortices, the flow wraps around the cube. Downstream of the cube, two symmetric structures are visible in the wake. Furthermore, vortex structures emerge from the top and side walls of the cube. Fig. 3 shows streamlines for u and w on horizontal crosssections at different vertical positions. The flow field is fully symmetric around the plane z* = 0; therefore, only region z* 6 0 is plotted. The horseshoe vortex upstream of the cube is clearly visible. On the lower cross-sections, the flow reverses directly upstream of the cube whereupon it bends around the obstacle. The two circulation structures downstream of the cube originate from the detached flow at the upstream edges of the cube and not from the horseshoe
3.1. Three-dimensional flow phenomena The occurring three-dimensional flow phenomena are analyzed in detail for Re = 150 and Ma = 0.1. To give a first impression of the flow, the three-dimensional iso-surfaces for k2 are presented in Fig. 2. Here, k2 is the second eigenvalue of X2 + S2, where X and S are the anti-symmetric and symmetric part of the velocity-gradient matrix [6].
Fig. 2. Top (top) and head (bottom) view of k2 iso-surface for Re = 150.
Fig. 3. Streamlines for Re = 150 and Ma = 0.1 at y* = 0.04, 1/4, 1/2 and 11/12 (from top to bottom).
A. van Dijk, H.C. de Lange / Computers & Fluids 36 (2007) 949–960
vortex. The horseshoe flow bends around the circulation areas and slightly expands again in the downstream wake behind the cube. It can be seen that the horseshoe vortex becomes thinner for larger y*. The horseshoe vortex has vanished at y* = 1/2, while the circulation zones behind the obstacle are still present. The structures emerging from the side walls of the obstacle and bending around it also remain present. At y ¼ 11 , the reversed flow at the back of the obstacle is still 12 visible. The side-wall structures no longer are closed vortices at this height. In Fig. 4, streamlines for u and v on four vertical crosssections (different z*-values) are displayed. Two prominent features can be recognized. First, the horseshoe vortex at the upstream corner of the channel floor and the obstacle is again clearly visible. Second, these cross-sections show the circulation zone downstream of the object. On the cross-section that coincides with the plane of symmetry (z* = 0), the center of the circulation zone is located behind the obstacle at approximately x* = 5 and y* = 0.8. The reversed flow is directed upwards near the top of the object, while a downward motion is found at the base of the obstacle. The attachment line is also visible, starting at x* ’ 6.4 for y* = 0. At z* = 1/4, the circulation zone no longer is a closed structure. At z* = 5/12, the circulation zone has disappeared, only the horseshoe vortex remains present. For 0.4 < y* < 1, the downstream flow moves upwards, past
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the circulation zone that is present near the obstacle center line. At z* = 7/12, directly next to the obstacle, the horseshoe vortex in front of the obstacle is shows as a slight upward motion directly above the channel floor. 3.2. Numerical accuracy To estimate the accuracy of the presented results, the calculation of the Re = 150 and Ma = 0.1 flow (as presented in the previous paragraph) has been repeated with double resolution in x, y or z direction. In Fig. 5 the streamlines at y* = 0.04 (near to the wall) resulting from two of these simulations can be found. In this figure the solution on the x-wise refined grid with 241 · 49 · 96 nodes (grid spacing: h/24 · h/24 · h/12) and z-wise refined with 121 · 49 · 192 nodes (grid spacing: h/12 · h/24 · h/24) are shown. These results show that the numerical resolution on the 121 · 49 · 96 grid is sufficient to capture all relevant flow features. To quantify the influence of grid refinement, the solution of the base (121 · 49 · 96) grid is compared to a number of more refined grids in Table 1. The first three columns in this table give the root-mean-square (rms) differences of u, v and w sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 RN s ðfr fs Þ2 rf ¼ ð28Þ U1 Ns
Fig. 4. Streamlines for Re = 150 and Ma = 0.1 at z* = 0, 1/4, 5/12 and 7/12 (from top to bottom).
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Fig. 5. Streamlines for Re = 150 and Ma = 0.1 at y* = 0.04 for the refined x-grid (top) and z-grid (bottom).
Table 1 The rms (ru, rv and rw defined in Eq. (28)) and the distance of the separation and attachment point to the obstacle (respectively xS and xR) for the different resolutions Grid
ru [103]
rv [103]
rw [103]
xS
xR
61 · 25 · 48 •121 · 49 · 96 241 · 49 · 96 121 · 97 · 96 121 · 49 · 192 241 · 49 · 192 241 · 97 · 192
39 10 21 9 9 5 –
9 3 4 3 2 1 –
10 3 5 3 4 1 –
0.95 1.04 1.06 1.04 1.02 1.023 1.021
2.79 2.46 2.57 2.44 2.37 2.407 2.405
The symbol • indicates the base grid, i.e. the grid used in the remainder of the paper.
where f is either u, v or w, the subscript r indicates the finest-grid (241 · 97 · 192) solution, while the subscript s indicates the solution on the studied grid. Ns is the total number of grid-points on the studied (s)-grid. Clearly, the rms-differences between the base-grid (121 · 49 · 96) solution and that of the more refined grid is small (one percent and less). The largest effect of refinement is on the main-stream velocity (u), while both other velocity components are predicted more accurately. Table 1 also shows that grid-refinement in x-direction only appears to decrease the accuracy of the solution. Furthermore, from the rms-values it follows that refinement in y- and z-direction has less influence. However, when refinement in x- and z-direction are performed simultaneously the solution closely resembles that of the finest grid. To show global grid convergence of the studied solutions the solution on a coarser grid (61 · 25 · 48) has been added. The rms-differences of this solution are in the order of 4% for u and 1% for v and w. Comparing this to the rmsdifferences of the base grid shows that the global conver-
gence of the solution is approximately quadratic, which means that the convergence is determined by the accuracy of the wall-point discretization. For the studied flow this is of course not a surprise, since all the flow structures, as described in the previous section, are in some way wallbounded. In the remainder of this paper, the separation point in front of the obstacle and the attachment point in its wake are being studied. These points will be defined as the far upstream point and the far downstream point on the centerline (y* = 0 and z* = 0) where (ou*/oy*) = 0. The distances between these points and the facing side of the block (xS for the upstream point and xR for the downstream point) are presented in Table 1 for all grids considered. This shows that the spacing extent of the horseshoe vortex (within 2%) and the wake (within 5%) barely depends on the chosen grid. Again, x- and z-refinement show the largest effect. It is worthwhile to mention that the effect of x- and z-refinement on xS and xR is opposite. For the x-wise refined grid both xS and xR increase, while for the z-wise refined grid they both decrease. When x-wise and z-wise refinement are performed simultaneously, the effects more or less cancel. Therefore, the accuracy of the predicted xS and xR is indeed in the order of 1–2%. The effect of y-wise refinement is much less. This also shows on the more refined level; when comparing the 241 · 49 · 192 and 241 · 97 · 192 grids only marginal differences are seen. To appreciate the accuracy of the solution it is worth remembering that the grid spacing on the base grid equals h/12 · h/24 · h/12. This means that the differences between xR and xS on the different grids (except for the coarsest grid) are well within the grid spacing. In the remainder of this paper, the chosen base grid (121 · 49 · 96 nodes, grid spacing: h/12 · h/24 · h/12) is assumed to provide sufficient resolution for accurately representing the flow. It is worthwhile noting that the changes in xS and xR due to the effect of Reynolds and Mach number (as they will be presented later in this paper) are considerably larger. 3.3. Effect of Reynolds number Fig. 6 shows the results for the u-w streamlines directly above the channel floor (y* = 0.04) Reynolds numbers ranging from Re = 50 to Re = 250. It can be seen that for higher Reynolds numbers, both upstream and downstream vortex structures increase in size. For Re = 50, the downstream circulation emerges from the upstream vortex. For Re = 100, it emerges from both the upstream vortex and the flow that is detached at the upstream obstacle edges. For Re = 150 and higher, the downstream circulation emerges from the detached flow at the obstacle edges. The horseshoe flow bends around these structures. For Re = 250, the second horseshoe vortex structure upstream of the first horseshoe is clearly visible. This second structure bends around the first horseshoe vortex and merges into it in the wake region.
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upstream of the object decays Re = 250. For Re = 50, ou oy monotonically from its value at the inlet up to a local minimum value at X* 2. For Re = 250, the derivative has three roots (ou*/oy* = 0) before it vanishes at the obstacle. The first dip corresponds with the small upstream vortex. Next, there is a small positive gradient where there exists a small secondary vortex in between of both primary horseshoe structures. The large dip in the velocity gradient close to the object corresponds with the major large horseshoe vortex. Downstream (x > 4) of the obstacle ou exhibits a local oy maximum followed by a local minimum in the case of Re = 250. The local maximum is present for the higher Reynolds numbers. It reflects the reversed flow structure at the downstream lower corner of the obstacle. For Re = 50 this local maximum is absent, implying that the reversed flow structure is not present. In Fig. 9 the correlation of xS and xR (as defined in Section 3.2) with the Reynolds number is presented. Both figures suggest a logarithmic correlation between the Reynolds number to the length of both zones. In our case (a closed channel) xR Re0.85. In [17] the length of the circulation flow of an obstacle on a flat plate in a boundary layer flow (with a boundary-layer thickness d equal to h) was studied numerically. They concluded that, for conditions similar to those presented here xR Re0.684. The slight difference between the study in [17] and our results may be due to the slightly different geometries (a closedchannel geometry in our study compared to the boundary layer in [17]). 3.5. Compressibility effect
Fig. 6. Streamlines (Ma = 0.1) at y* = 0.04 for Re = 50, 100, 200 and 250 (from top to bottom).
Fig. 7 shows, like Fig. 6, that the upstream and downstream vortices grow in size with increasing Reynolds number. The second horseshoe vortex, situated upstream of the first one, is clearly visible for Re = 250. The vortex structure downstream of the upper half of the obstacle is no longer closed for Re = 250. For Re = 50 and Re = 100 the downward flow, as found for Re = 150 and higher, is no longer present at the downstream junction of the channel floor and the obstacle. 3.4. Separation and attachment points The growing number of structures for increasing Reynolds number is also reflected by the ou*/oy*-derivative (with u u=ub ) on the channel floor at the z-plane center line. In Fig. 8 this derivative is given upstream (x < 3) and downstream (x > 4) of the object for Re = 50 and
To judge whether the flow depends on the Mach number for subsonic conditions, the effect of compressibility is evaluated by comparing the results at Re = 150 and Ma = 0.1 with those for Ma = 0.5 and Ma = 0.6. The results of these simulations are presented in Figs. 10 and 11. From the streamlines close to the channel floor, as presented in Fig. 10, it is clear that the results for Ma = 0.1 and 0.3 are very similar. There appears to be a small change of the flow pattern just ahead of the block. The vortical structure shows a slightly stronger downward motion close to the leading edge. Moreover, the secondary motion at the top of the horseshoe vortex seems to increase with the Mach number. These observations lead to the conclusion that the horseshoe vortex may become somewhat stronger with increasing Mach number. When the results for Ma = 0.6 are taken into consideration, the growing strength of the vortex with increasing Mach numbers becomes more obvious. The reason for this growth may lie in the temperature in the stagnation point. As the Mach number increases, so does the stagnation temperature. Now, if we would assume that the compression ahead of the block is approximately isentropic, we can show that the kinematic viscosity (m = l/q) in the horseshoe vortex must decrease with increasing Mach number. This leads
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Fig. 7. Streamlines (Ma = 0.1) at z* = 0 for Re = 50, 100, 200 and 250 (from top to bottom).
to reduced diffusion, which induces an increased vortex strength. Another striking change can be seen in the downstream region (in Fig. 10). The higher Mach number flow clearly exhibits a reduced distance xR to the downstream attachment point. Clearly, the circulation area behind the block becomes more compact, indicating an increased lateral flow. The flow is obviously sucked in from the sides due to the increased dynamic pressure. Both flow features are also shown in Fig. 11. This figure gives the streamlines on the centerplane z* = 0. Again, for Ma = 0.3 and 0.6 the backflow is found to be somewhat larger indicating an increased strength of the vortex. The contours also show that the spatial extent of the influence of the block expands further in upstream direction. On the other hand, in the downstream region the wake narrows. When the influence of the Mach number on xS and xR is studied, we find the results as depicted in Fig. 12. It is clear that the effect of the Mach number on both the separation and attachment point is much less than that of the Reynolds number. For the considered Mach numbers, the (downward) compression of the horseshoe vortex leads to a slight (20%) increase of the length of the separation zone from xS 1.04 at Ma = 0.1 to xS 1.23 at Ma = 0.6. As is apparent from Fig. 11 the attachment point moves closer to
the block with increasing Mach number. This leads to a 10% (linear) decrease of xR from xR 2.46 at Ma = 0.1 to xR 2.26 at Ma = 0.6. It is worthwhile noting that, although the Mach number has only a marginal effect on xS and xR, the changes of 10–20% are still well beyond the numerical accuracy (1–2%). Fig. 13 gives the streamwise velocity on the obstacle centerline at y* = 0.04 in front (top) and behind (bottom) the obstacle for increasing Mach numbers (0.1, 0.3 and 0.6). As can be seen from Fig. 13 (top) the increase of xS with higher Mach numbers goes hand in hand with the appearance of the ‘double’ vortex structure; just as in the case of the increasing Reynolds number. The profile behind the object (Fig. 13, bottom) shows that the velocity increases much faster for higher Mach numbers. This, again, indicates that due to the increased dynamical pressure there is a larger lateral inflow of fluid, thus leading to a faster acceleration. 3.6. Validation of the results In [2] the flow around a square cylinder mounted on a flat plate has been studied experimentally for Re = 1000 (based on a uniform inflow velocity and the cylinder diameter). Smoke visualizations and Particle Image Velocimetry measurements are adopted. The main differences between their set-up and the numerical set-up presented in this
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paper is their uniform inflow velocity profile and the absence of a top channel wall. Furthermore, the square cylinder height equals 0.3 times its diameter, in contrast to the cubic geometry in our study. For higher Reynolds numbers, the flow field around several prismatic obstacles has been experimentally investigated in e.g. [16]. These experiments are performed in fully developed channel flow. The Reynolds number, based on the height of the channel and the mean bulk velocity, lies between 8 · 104 and 1.2 · 105. Several distinct similarities can be observed between these experimental results and our numerical results. Similar to the structures visible in Fig. 2, the visualization studies show one large and one smaller horseshoe vortex structure. Furthermore, for both [2] and our numerical results, a small reversed flow structure (leading to the bump in ou/oy in Fig. 8 for Re = 250) is present at the channel floor. In [22] the laminar flow between two parallel plates with a cube mounted on the lower wall is numerically investigated. The Reynolds number, based on the bulk mean velocity and the obstacle height, is varied from Re = 5 to Re = 1500. The length of each side of the cubic obstacle
is fixed as one half of the channel height (h/H = 1/2). The streamwise and spanwise domain lengths equal 10h and 7h, respectively. For Reynolds numbers Re > 250, the streamwise domain length is increased by 5h. Periodic boundary conditions are applied in spanwise direction. At all solid walls, no-slip boundary conditions is imposed. A parabolic velocity profile is prescribed at the inlet. In [22] a clear drawing of the u and w streamlines directly above the channel floor is presented for Re = 350. It shows a steady symmetric flow profile around the z-plane centerline. The inward-bending flow at the cube edges and the horseshoe vortex upstream of the cubic obstacle are also present similar to those visible in Fig. 6. A striking difference between the results in [22] and those of this study is the topology of the wake. In [22] the two downstream circulation structures cannot be identified. It must be noted that our results are more in line with the experimental results by [2] where the two contra-rotating vortices with vertical axis remain confined behind the obstacle have also been observed. Furthermore, our wake results bear a striking resemblance with the close-to-the-wall results presented in [4]. In the latter numerical study large eddy simulations are performed on the film-cooling problem. Although the
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parameters to that problem are somewhat different to ours, the wake region in [4] also shows the closed double vortex system, which characterizes the wake as we find it. In [22] the relation between the Reynolds number and the distance between the far upstream separation point and the upstream face of the obstacle (xS) is presented. Their results are compared with the present simulations in Fig. 14. A notable feature in this figure is the jump found in the results of [22] at Re = 350. In [22] this is attributed to the changing nature of the flow at Re = 350. However, the jump also coincides with an elongation of their computational domain by a factor 1.5. Our results seem to correspond very well to those of the elongated domain in [22]. This might suggest that for the shorter domain the outflow boundary conditions used in [22] still have some influence on the attachment point. 4. Conclusions
Fig. 10. Streamlines (Re = 150) at y* = 0.04 for Ma = 0.1, 0.3 and 0.6 (from top to bottom).
The subsonic laminar flow around a cubic obstacle placed on the floor of a spanwise periodic channel has been studied numerically. As a first step a base solution is described, it consists of a flow at Re = 150 and Ma = 0.1 solved on a grid of 121 · 49 · 96 nodes. Four main flow zones are identified: (i) a horseshoe-vortex system, (ii) inward bending flow at the side walls of the obstacle, (iii) a vortex with a horizontal axis at the downstream upper half of the obstacle and (iv) a downstream wake containing two counter-rotating vortices with vertical axes. When the solution of the base grid (121 · 49 · 96 nodes) is compared to a refined grid (241 · 97 · 192 nodes) the length of the
Fig. 11. Streamlines (Re = 150) at z* = 0 for Ma = 0.1, 0.3 and 0.6 (from top to bottom).
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separation and circulation zones are predicted with an accuracy of 2% and 5%, respectively. Starting from the base solution, the effect of the Reynolds number and the Mach number is studied. The Reynolds number is increased in four steps from Re = 50 to Re = 250 with the Mach number fixed at Ma = 0.1. Next, the Mach number is increased to Ma = 0.3 and Ma = 0.6 with a fixed Reynolds number at Re = 150. For all cases investigated, a steady flow field is found. For Reynolds numbers Re P 150, three additional flow structures are formed: (i) an additional horseshoe vortex, (ii) a vortex with a horizontal axis on top of the obstacle and (iii) a vortex at the downstream junction of the obstacle and the channel floor. The latter structure counterrotates the vortex at the downstream upper half of the obstacle. For Re = 250, the vortex at the downstream upper half of the obstacle no longer forms a closed structure. A logarithmic correlation is found between the Reynolds number and the distance between the block and the far upstream separation point (xS) and the far downstream attachment point (xR).
Fig. 13. u* above the plate at the centerline (Re = 150) for Ma = 0.1, 0.3 and 0.6 in front (top) and behind (bottom) the object.
x
Fig. 12. Correlation of xS (top) and xR (bottom) with the Mach number (Re = 150).
1
0.6
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When increasing the Mach number it becomes obvious that (at these subsonic conditions) the flow pattern is highly Re-dominated. There are however, some differences detected. The major points are an increase in the strength of the horseshoe vortex, an increased upstream influence of the obstruction and a decreased length of the circulation zone. It appears that the latter two are caused by the increase of the pressure differences due to the increased dynamic pressure. The slight increase in the strength of the horseshoe vortex is attributed to the local decrease in kinematic viscosity. References [1] Bosma AJ, Stoffels GGM, van der Meer TH. Numerical simulation of heat transfer of nickel foam in thermo-acoustic waves. In: Proceedings of the EuroTherm74: Heat transfer in unsteady and transitional flows, 2003. pp. 79–84. [2] Calluaud D, David L, Texier A. Study of the laminar flow around a square cylinder, Ninth international symposium on flow visualization, 2000; paper 402. pp. 6. [3] Dimaczek G, Kessler R, Martinuzzi R, Tropea C. The flow over twodimensional, surface mounted obstacles at high Reynolds numbers. In: Seventh symposium on turbulent shear flows. vol. 10(1), 1989. pp. 1–6. [4] Guo X, Schro¨der W, Meinke M. Large-eddy simulations of filmcooling flows. Comput Fluids 2006;35:587–606. [5] Hussein HJ, Martinuzzi RJ. Energy balance for turbulent flow around a surface mounted cube placed in a channel. Phys Fluids 1996;8:764–80. [6] Jeong J, Hussain F. On the identification of a vortex. J Fluid Mech 1995;285:69–94. [7] Jovanovic MB, de Lange HC, van Steenhoven AA. Influence of laser drilling imperfection on film cooling performances, ASME paper, 2005; GT2005-68251. [8] Jovanovic MB, de Lange HC, van Steenhoven AA. Influence of hole imperfection on jet-cross flow interaction. Int J Heat Fluid Flow 2006;27:42–53.
[9] Kennedy CA, Carpenter MH, Lewis RM. Low-storage, explicit Runge–Kutta schemes for compressible Navier–Stokes equations. Appl Num Math 2000;35:177–219. [10] Korichi A, Oufer L. Numerical heat transfer in a rectangular channel with mounted obstacles on upper and lower walls. Int J Therm Sci 2005;44:644–55. [11] Krajnovic S, Davidson LV. Flow around a three-dimensional bluff body. In: Ninth international symposium on flow visualization, 2000; paper 177. pp. 10. [12] de Lange HC. Split time-integration for Low-Mach Number compressible flows. Comm Numer Meth Eng 2004;20(7):501–9. [13] de Lange HC. Inviscid flow modelling using asymmetric implicit finite difference schemes. Int J Numer Meth Fluids 2005;49: 1033–51. [14] Lele SK. Compact finite difference schemes with spectral-like resolution. J Comput Phys 1992;103:16–42. [15] Martinuzzi R, Melling A, Tropea C. Reynolds stress field for the turbulent flow around a surface-mounted cube placed in a channel. In: Ninth symposium on turbulent shear flows, 1993. 13-4-1. [16] Martinuzzi R, Tropea C. The flow around surface-mounted, prismatic obstacles placed in fully developed channel flow. J Fluids Eng 1993;115:85–91. [17] Ngo Boum GB, Martemianov S, Alemany A. Computational study of laminar flow and mass transfer around a surface-mounted obstacle. Int J Heat Mass Transfer 1999;42:2849–61. [18] Poinsot TJ, Lele SK. Boundary conditions for direct simulations of compressible viscous flows. J Comput Phys 1992;101:104–29. [19] Roget A, Brazier JP, Cousteix J, Mauss J. A contribution to the physical analysis of separated flows past three-dimensional humps. Eur J Mech B/Fluids 1998;17:307–29. [20] Sesterhenn J. A characteristic-type formulation of the Navier–Stokes equations for high order upwind schemes. Comput Fluids 2001;30: 37–67. [21] Tanaeva I. Low-temperature cryocooling. PhD-thesis, Technische Universiteit Eindhoven, 2004. [22] Yang KS, Hwang JY. Numerical study of horseshoe vortices around a cubic obstacle in a channel, ASME 2000 Fluids Engineering Division Summer Meeting, 2000; FEDSM00-11236. [23] Xiaolin Zhong. High-order finite-difference schemes for numerical simulation of hypersonic boundary layer transition. J Comput Phys 1998;144:662–709.