Simulation of turbulent flow over a backward-facing step at Re=9000
Simulation of turbulent flow over a backward-facing step at Re=9000 Michal A. Kopera,1, a) Robert M. Kerr,2 Hugh Blackburn,3 and Dwight Barkley4 1)
Department of Earth and Planetary Sciences, University of California, Santa Cruz, USA 2) Mathematics Institute, University of Warwick, United Kingdom 3) Department of Mechanical and Aerospace Engineering, Monash University, Australia 4) Mathematics Institute, University of Warwick, Coventry CV4 7AL, United Kingdom (Dated: 11 October 2017)
Turbulent flow in a channel with a sudden expansion is simulated using the incompressible Navier-Stokes equations. The expansion ratio is ER = 2.0 and the Reynolds number, based on the step height and mean inlet velocity, is Reh = 9000. The discretization is performed using a spanwise periodic spectral/hp element method. The turbulent inflow is regenerated from a channel segment upstream of the step. The investigation of pressure and streamwise velocity fluctuations indicates that the interaction of vortices with the recirculation bubble is responsible for the flapping of the reattachment position, which has a characteristic frequency of St = 0.078. Time and spanwise averages show secondary and tertiary corner eddies in addition to the primary recirculation bubble. Streamlines show an additional small eddy forming at the downstream tip of the secondary corner eddy. This eddy has the same circulation direction as the secondary vortex. Analysis of three-dimensional time-averages shows a wavy spanwise structure that leads to spanwise variations of the mean reattachment location. Keywords: backward facing step, turbulent flow, direct numerical simulation I.
A.
INTRODUCTION
The flow over a backward-facing step (BFS) is a prototype for separating, recirculating and reattaching flow in nature and numerous engineering applications. Examples include the flows around buildings, inside combustors, industrial ducts and in the cooling of electronic devices. All of those cases share one common feature: That an adverse pressure gradient (usually due to a sudden change of geometry) causes the boundary layer to separate from the surface and form a mixing layer, which eventually re-attaches to the surface. The presence of separation, recirculation and reattachment drastically changes the transport of momentum and heat. The backward-facing step is a prototype of these scenarios, as it demonstrates the phenomena with a simple geometry, one that is easy to set-up experimentally, as well as model computationally. Recent publications regarding direct numerical simulations (DNS) of turbulent flow over a BFS include Le, Moin, and Kim 1 for a step in an open channel, and Meri and Wengle 2 and Barri et al. 3 for an expansion in a closed channel. Recently Lamballais 4 studied the effect of rotation on flow in a rotating channel with a sudden double expansion. The Reynolds numbers ranged between 3300 − 5600. In this work, we are providing statistical and instantaneous data for Reh = 9000, with some analysis regarding the flow structure.
a) Electronic
mail: http://www.koperalab.ucsc.edu
[email protected].;
Survey of previous work
BFS flow has been investigated experimentally many times. Early work with expansions on one or both walls was reviewed by Abbot and Kline 5 , and an extensive overview of experiments on recirculating flows in different configurations performed up to 1970 was provided by Bradshaw and Wong 6 . Since then there have been a number of experimental studies that examined similar configurations with expansions7–15 . The first two-dimensional simulations addressed only the mean flow over a BFS for either the laminar9 or the transitional16 regime. Coherent structures were first identified17 in both two and three dimensions by using a prescribed inlet velocity profile and superimposed noise applied to direct and large-eddy simulations. True comparisons with experimental statistics began with the Le, Moin, and Kim 1 simulations of a BFS in an open channel. They used an expansion ratio of ER = 1.2 at Re = U0 H/ν = 5100 (with Uo as maximum inflow velocity) and obtained a characteristic frequency of reattachment of St = 0.06, a mean reattachment length of Xr = 6.28, and the instantaneous reattachment location varied in the spanwise direction, all in agreement with a concurrent experimental study18 . Le, Moin, and Kim 1 (LMK) became a reference for all turbulence models and set a standard in separated turbulent flow simulations by providing the budgets of all Reynolds stress components, and reporting averaged velocity and pressure fields. This paper documents many similarities between the properties of the two simulations, LMK and here, despite differences in the Reynolds numbers, nature of the inflow and whether it is an open or closed channel. The paper extends the simulation
Simulation of turbulent flow over a backward-facing step at Re=9000 database of turbulent backward-facing flows to higher Reynolds numbers, as well as provides an analysis of the fluctuations underlying the oscillations of the reattachment line in terms of velocities, vortex structures, wall shear stresses and frequency spectra. The goal is to provide a more complete picture of the origins of the reattachment oscillations, a picture that can then be applied to other flows with oscillatory behavior. This paper is organized as follows: Section II presents the governing equations, boundary conditions and numerical methods used for the simulations. Section III provides a discussion of the results, including the inlet velocity profiles, reattachment length, as well as averaged velocity and pressure fields. The data is validated by comparison with existing experimental and numerical results. The investigation of the dynamic behavior of the reattachment position and characteristic flow frequencies is followed by the analysis of average flow fields, which provides an insight into the structure of the flow. The paper is concluded in Section IV.
II.
METHODS AND SIMULATION SET-UP
A.
Governing equations and numerical method
The flow over a backward-facing step is governed by the incompressible Navier-Stokes momentum equation (1a) with an incompressibility constraint (1b): ∂u 1 + (u · ∇)u = − ∇p + ν∆u ∂t ρ ∇ · u = 0,
(1a) (1b)
where u = {ui } = [u, v, w] is the velocity vector, ρ is a constant density, p is pressure and ν is kinematic viscosity. Equations (1) were solved using a spectral element code19 , which uses a two-dimensional spectral/hp element method (SEM) in x and y plane, and Fourier transform in z for spatial discretization. For time integration the code uses the stiffly-stable multi-step velocitycorrection method20 . SEM is stabilized using a low-pass Boyd-Vandeven filter21 .
B.
Geometry and boundary conditions
The geometry of the domain is shown in Fig. 1. The coordinate frame defined by the (x, y, z) axes, where x indicates the streamwise, y the vertical and z the spanwise direction. Its origin is positioned at the bottom of the step. The length scale is defined by the step size h. The inlet channel is Li = 12h long, and the outlet channel has dimensions Lx = 29h and Ly = 2h, giving an expansion ratio of ER = Ly /(Ly − h) = 2. The boundaries confining the channel from top and bottom are modeled as no-slip, no-penetration walls.
2
In the spanwise direction, we apply a periodic boundary condition. Le 22 reports that a periodic length of Lz = 4h was adequate to tail off the two-point correlations for u, v and w near the wall, while away from the wall in the free shear layer some correlations remained at approximately 0.1. One reason is the presence of spanwise rollers in the free shear layer. Here we use the periodic length of Lz = 2πh to address this issue. At the inlet boundary, we provide a turbulent inflow by regenerating a velocity profile from a plane downstream at x = −4h in the inlet channel. Previous studies3,23 used a similar method, where an auxiliary channel flow simulation was used to generate turbulent inflow. Here we embed the auxiliary simulation in the main simulation by forming a regeneration area between x = −12h and x = −4h. The same approach was used recently by Lamballais 4 . To keep the mass flow constant, at the end of each time step the velocity in the entire domain was modified using a Green’s function ug , which is a solution to the Stoke’s equation with the same boundary conditions as the original problem. The modified velocity is u ← u + α(t)ug , where α(t) is a time dependent correction parameter computed at each time-step to obtain desired mass flow rate24 . A conventional no-stress outlet boundary condition ∇u · n |∂Ωo = 0 was prescribed at the outlet plane x = 29h.
C.
Grid resolution
The two-dimensional SEM used in this work expands the solution into quadrilateral elements of orthogonal polynomials, which pave the entire computational domain. The mesh in the inlet channel has eleven elements in the vertical direction, with vertices locations following the Chebyshev points distribution. The element closest to the wall has dimension ∆ye+ ≈ 16, based on the viscous length ν/uτ , where the friction velocity uτ = 0.061Ub is measured at x = −8h in the inlet channel. With eleven Legendre-Gauss-Lobato points in each direction inside an element, the first point away from the wall is located at ∆y + = 0.528 with seven points below y + = 10. In the streamwise direction the element sizes vary from ∆x+ e ≈ 136 near the inlet (effective resolution within + element ∆x+ ≈ 13.6) to ∆x+ e ≈ 27 at the step (∆x ≈ 2.7). The element sizes slowly increase as they approach the outflow (see Fig. 1). A single x−y slice of the domain consists of 2845 2D elements and 344245 nodal points. In the spanwise direction the number of collocation points was Nz = 128, which corresponds to 64 Fourier modes and ∆z + = 27. For this study, we define the effective resolution to be the average distance between nodal points within the element, computed as the size of the element divided by the 1 number of points: ∆e = (∆xe ∆ye ) 2 /NP , where ∆xe and ∆ye are the streamwise and vertical sizes of the element for which the effective resolution is computed. Fig. 2 compares the effective resolution with Kolmogorov scale,
Simulation of turbulent flow over a backward-facing step at Re=9000
3
FIG. 1. Overview of domain geometry and mesh. The inlet channel has dimensions Li × h × Lz , the channel after expansion has dimensions Lx × Ly × Lz , with Lx = 29h, Ly = 2h and Lz = 2πh.
FIG. 2. Spanwise averaged grid spacing ∆e divided by the locally estimated Kolmogorov scale ηK .
the HECToR XT4 system using 64 processors.
estimated by ηK =
1 3 4
ν
,
(2)
where = 2νSij Sij is the local energy dissipation rate and Sij represents the rate-of-strain tensor. The effective resolution does not exceed 7ηK and for most of the domain it is below 5ηK . The wall regions are very well resolved with the effective resolution below 3ηK . To verify the resolution in the spanwise direction, the modal energy decay in several places in the flow was examined. Fig. 3a shows the result obtained in the inlet channel and panels b - d present the results for different places in the shear layer. For all of the components at the x − y positions (except for Evv in panel d, as discussed in Sec. III E 4), there are clear drops in the modal energy over at least two decades, indicating that there is an adequate spanwise resolution in the shear layer. D.
Simulation parameters
This simulation used Reynolds number Reh = 9000, based on the step height and mean inlet velocity. Using inlet channel friction velocity uτ as velocity scale gives Reτ = 550. For comparison with channel flow simulations, one should use the inlet channel half-width δ = h/2 for length scale, giving Reδτ = 275 and Reδ = 4500. The simulation consisted of ≈ 4.4e7 nodal points, and required time-step size of ∆t = 5e−4 h/Ub . It was run on
III. A.
RESULTS Inlet
We compared the results with channel flow simulations in the literature25,26 to validate the turbulent inflow used in the BFS simulation. Figure 4(a) shows the averaged U velocity profile. The results show the same slope in the log layer as LMK1 , which is only slightly different from the channel flow profiles. Current simulation profile is closer to Reδτ = 395 result of Moser, Kim, and Mansour 26 than Reδτ = 18025 indicating that the Reynolds number is high enough for the flow not to exhibit low Reynolds number effects. The agreement between LMK1 and current simulation may also indicate, that it the presence of the step affects the inlet boundary layer profile more than low Re effects, which should be present in LMK1 results. Turbulent intensity profiles in Fig. 4(b) are in good qualitative agreement with the reference data26 . There are some noticeable differences, particularly in the peak of 0 profiles due to the difference in Reτ . u0rms and wrms Apart from the statistical properties of the flow in the inlet section, another concern was the influence of the length of the inlet channel and the role of the inlet regeneration on the dynamics of the flow. The power spectrum in Fig. 5 shows that the periodic regeneration region introduces an artificial frequency St = 0.127, and its
Simulation of turbulent flow over a backward-facing step at Re=9000 (a)
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FIG. 3. Time averaged spectrum of u0 (solid line), v 0 (dashed line) and w0 (dot-dashed line) energy at (a) (x, y) = (−2, 1.5), (b) (x, y) = (0.1, 1), (c) (x, y) = (4, 1), (d) (x, y) = (4, 0.01).
harmonics, which corresponds to the periodic area length of Lr = 8h, assuming that the frequency is computed as fr = Ub /Lr , and St = f h/Ub . While this mode could have been suppressed by increasing the length of the recirculation zone, after a thorough examination of spectra at different locations in the flow it was decided that the recycling frequency does not have any significant influence on the dynamics discussed in Sec. III D.
Reattachment length and coefficient of friction
The reattachment length Xr is defined as an average distance of the flow reattachment position from the step. We diagnose it from the time and spanwise average coefficient of friction (Cf ) at the bottom wall by taking Xr as the location of Cf = 0. Table I summarizes the Xr values for a number of computational and experimental studies, along with the peak negative Cf , its position downstream from the step, the expansion ratio ER and
0.1
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FIG. 4. (a) Inlet velocity mean U and (b) fluctuations u0rms 0 0 (solid line), vrms (dashed line), wrms (dot dashed line) profiles time and spanwise averaged at x = −2.0. The statistics were collected over an averaging time of Tave ≈ 200h/Ub . Profiles are compared with results of turbulent channel flow DNS simulations of Kim, Moin, and Moser 25 (circles, Reτ = 180), Moser, Kim, and Mansour 26 (crosses, Reτ = 395) as well as the backward-facing step simulation of Le 22 (squares). Solid line in panel (a) denotes current simulation (Reτ = 226 using inlet channel half-width for length-scale).
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FIG. 5. Power spectrum of u0 - power spectral density of spanwise averaged u0 velocity fluctuation at x = −2.0, y = 1.5. The periodic regeneration area introduces an artificial frequency St = 0.127, and its harmonics, which corresponds to the regeneration region length of 8h.
Simulation of turbulent flow over a backward-facing step at Re=9000 Remarksa N, C, 3D N, C, 3D N, O, 3D N, O, 2D E, C E, C E, O E, O E, O
Case current Barri et al. 3 Le, Moin, and Kim 1 Armaly et al. 9 Kasagi and Matsunaga 12 Adams and Johnston 10 Jovic and Driver 11 Spazzini et al. 13 Chandrsuda and Bradshaw 27 a b c
Reh 9000 5600 4250b 4000b 4834b 30000b 8700b 8300b 100000
5
Xr 8.62 7.1 6.28 8.0 6.51 6.3 5.35 5.39 6.0
Cf,min c −2.9 · 10−3 −2.6 · 10−3 −2.89 · 10−3 −0.885 · 10−3 −2.0 · 10−3 −1.87 · 10−3 -
ER 2.0 2.0 1.2 2.0 1.5 1.25 1.09 1.25 1.4
X(Cf,min ) /Xr 0.62 0.54 0.61 0.63 0.63 0.6 -
N/E - numerical/experimental result; C/O - closed/open channel, 2D/3D - two or three-dimensional simulation The values of Re scaled using Ub . Cf is scaled using maximum inlet velocity U0 , assuming U0 = 1.22Ub based on an average velocity profile for a turbulent channel flow.
18 16 14 12 10 8 6 4
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TABLE I. The summary of previous numerical and experimental works on backward-facing step geometry, with reattachment length Xr , expansion ratio ER , coefficient of friction minimum Cf,min and its location X(Cf,min ) .
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FIG. 6. Reattachment length Xr as a function of Reynolds number obtained from Armaly et al. 9 (circles), Barri et al. 3 (triangle) and current simulation (square).
Reynolds number. In Fig. 6 we select only the results with ER = 2 for better comparison. Armaly et al. 9 showed that Xr depends strongly on Re in the laminar and transitional regime (with Reh = 3300 identified as the lower limit of the turbulent flow), but with no Reynolds number dependence in the turbulent regime. A Recent simulation by Barri et al. 3 and current result extend the range of Re and indicates that some Reynolds number dependence still exists, however much weaker than for laminar and transitional regimes. Similar conclusions come from the comparison of results of Spazzini et al. 13 and Adams and Johnston 10 , where tripled Re causes ≈ 17% increase in the reattachment length. To compare with previous results, we plot in Fig. 7 the coefficient of friction against the x coordinate normalized by the reattachment length. Note that for this comparison we scale the coefficient of friction using maximum centerline velocity, rather than Ub . The relative proximity of the minima of the coefficients of friction between the simulations (lines) was a surprise. Owing to significantly increased Reynolds number, one would have expected the minima of Cf to decrease. The fact that LMK1 has no top wall does not seem to affect the coef-
0
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FIG. 7. Coefficient of friction as a function of distance from the step, normalized with the reattachment length. Our Cf (solid line) is compared with results of simulations of Le, Moin, and Kim 1 (dashed line), Barri et al. 3 (dot-dashed line) and experiments by Adams and Johnston 10 (triangles), Jovic and Driver 11 (circles) and Spazzini et al. 13 (squares).
ficient of friction at the bottom wall in the recirculation area. It might, however, have an impact on the regeneration of the flow, as there seems to be a discrepancy between LMK1 and other results downstream of the reattachment position.
C.
Wall shear stress
Figure 8 shows the time-average of wall shear stress on the bottom wall. There is a clear variation of the reattachment line in the spanwise direction. The dot-dashed line marks the spanwise averaged reattachment length Xr = 8.62h. Closer to the step, there is another region of forward flow where τw > 0 (warm colors), indicating the presence of a secondary recirculation bubble, which is also not homogenous in the spanwise direction. Closer examination also reveals a small area of negative τw nearest to the step. We discuss this sequence of recirculation
Simulation of turbulent flow over a backward-facing step at Re=9000
FIG. 8. Time-average of shear stress at the bottom wall. Solid black line marks τw = 0 and separates the regions of forward and reversed flow.
bubbles in Sec. III E 2. Instantaneous streamwise component of the wall shear stress is shown in the Fig. 9, with dashed line marking Xr = 8.62h. Unlike Fig. 8, here we do not see a continuous reattachment position indicated by τw = 0. In this snapshot, four main regimes can be defined: forward flow for x > 12h, mixed flow in the reattachment zone for 6 < x < 12, reversed flow for 2.5 < x < 6 and the secondary bubble with forward flow near the wall for x < 2.5. Between x = 6h and x = 9h we observe three forward/reverse pairs in the spanwise direction, which we believe are the boundary layer signatures of streamwise persistent structures, which are discussed in Sec. III E 4 in more detail. In the movie (as well as in a sequence of images in Kopera 24 (pg. 116)), it can be seen that at as the reverse flow area moves downstream there is an increase of the instantaneous reattachment length Xr . At t = 65.0h/Ub the three streaks of forward flow start to merge into a larger spanwise structure that starts to cutoff a zone of reverse flow between x = 7.5h and x = 9.0h. This enclosed reverse flow zone moves downstream and disappears at around t = 70.0h/Ub . The complex mixture of forward and reverse flow patches in the secondary bubble can also be observed without any clear correlation between the behavior of the main reattachment location and secondary bubble.
D.
Oscillations of the reattachment position
The complex behavior of the reattachment position can be presented as a time evolution of spanwise averaged wall shear stress, presented in Fig. 10. The solid line denotes separation between forward (white) and reverse (gray) flow. The temporal variations in the reattachment length form an oscillating pattern. First, the reattachment length increases slowly in a roughly linear fashion with an average slope of 0.3Ub (i.e. t = 58 − 65h/Ub ). At some point, a forward flow region forms upstream of the current reattachment (t=65h/Ub ). This forward flow zone eventually overtakes the downstream reverse flow zone, thus closing the leaning saw-tooth shape, as at t = 70h/Ub . Simultaneously the upstream limit of the
6
new forward flow area becomes the new reattachment position. This oscillating pattern is not very regular and carries small-scale structures on top of it. The secondary bubble lacks the small scale structure of the main recirculation zone, yet exhibits a similar but inverse saw-tooth shape, with a negative slope of roughly −0.08Ub . LMK1 reported a similar saw-tooth shape to the Xr t plot for Reh = 4250 (originally Re = 5100 based on U0 ) and ER = 1.2. Another similarity between the two cases is the frequency of the oscillations. In both figures, there are approximately eight saw-tooth shapes in 100h/Ub period. Oscillatory behavior of the reattachment was also reported by Schafer, Breuer, and Durst 28 . Fig. 11 provides the quantitative analysis of those oscillations using the frequency spectra of the streamwise velocity fluctuations u0 (panel a) and the pressure fluctuations p0 (panel b) for different locations near the reattachment position. The positions of the selected measurements points are depicted on the top plot, and the full set is available in Kopera 24 . For each (x, y) location the spectrum was computed for all z planes using 8000 samples taken every 50 time steps. The spectra were then averaged over the spanwise direction. In Fig. 5 we have found that the inflow was dominated by the frequency corresponding to the length of the periodic regeneration (St = 0.127), and its subharmonics. Spectra near the step edge (point #2) show only a slight peak at the regeneration frequency, both for the velocity and pressure fluctuations. In the mixing layer (point #5) there is a broader band of frequency peaks, with St = 0.127 present in the pressure plot but not in the velocity fluctuations. In the reattachment region, however, the most dominant frequency in both velocity and pressure fluctuations is St = 0.078, with the regeneration frequency showing only weakly in the pressure spectrum. There is no evidence of that frequency in the inlet channel, but a similar frequency of St = 0.068 shows up in the mixing layer. This could be either a harmonic of St = 0.127, or a natural frequency of the flow. The latter conclusion is supported by the results in the literature, which report St ≈ 0.06 frequency of the reattachment flapping1 and St = 0.08 for large Kelvin-Helmholtz structures in the mixing layer17 . The fact that the same dominating frequency is present in both pressure and velocity fluctuations indicates that the presence of a vortex and behavior of the reattachment position are correlated and tuned to a characteristic frequency of St ≈ 0.068 − 0.078. To further exclude the influence of the regeneration frequency on the reattachment oscillations, an additional simulation with shorter regeneration length (5h) was performed. The spanwise length was set to Lz = 0.75π and the spanwise resolution Nz = 48. Fig. 12 presents the spectra taken in the inlet channel (point #1) and reattachment zone (point #9) of this additional simulation. It clearly shows the new regeneration frequency of St = 0.219 and its harmonics in the inflow. The characteristic frequency in the reattachment zone is St = 0.068, which clearly is not related to the new regeneration fre-
Simulation of turbulent flow over a backward-facing step at Re=9000
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FIG. 9. Instantaneous streamwise component of the wall shear stress (wall traction) at the bottom wall. Solid black lines marks τw = 0 and separate the regions of forward (warm colors) and reversed (cold colors) flow. Dashed red line marks the space-time average reattachment location. (Multimedia view)
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FIG. 10. A space-time plot of spanwise-averaged contours of the instantaneous streamwise component of the shear stress at the bottom wall. Solid black line marks τw = 0 and separates the regions of forward (white) and reversed (grey) flow.
quency and confirms the finding that St = 0.068 − 0.078 is indeed a characteristic frequency of the recirculation and reattachment zone. Lamballais 4 conducted a thorough investigation of the recycling technique, and found that the regeneration length 6.25h did not significantly affect the main time scales of the flow. No correlation was found between the velocity fluctuations in the near-wall region of the inlet channel, indicating that chosen recycling length was sufficient to avoid the introduction of any residual time periodicity. E.
Averaged flow field
The flow data was averaged over the time Tave = 200h/Ub using 8000 samples. The averaging was initiated after an initial burn-in time of TBI = 50h/Ub required for the decay of the initial transients from the initial condition, obtained from an initial simulation with lower Re. The length of the burn-in process was based on the time required to adjust the streamwise component of viscous force on walls Z Fτ x = τxj nj dW
at the change in the time and spanwise averaged reattachment length. Xr remained within 0.1% of the final value during the last 25h/Ub (equal to about one flowthrough time) and bounded by ±0.4% limit in the last four flow-through times. This provides a reasonable level of confidence in the convergence of the collected statistics. However, this simulation time was not enough to obtain a completely spanwise homogeneous time-averaged flow field, as visible in Fig. 8 In the following analysis, we define U , V , W to be the time and spanwise average values of velocity components in the x, y and z directions respectively and P the average pressure. 1.
Pressure field
We report pressure in terms of the static pressure coefficient, defined as CP =
P − P0 , 1 2 2 ρUb
where P0 is a reference pressure taken at x = −4h, y = 1.5h. In Fig. 13 we plot the maximum pressure coefficient against the ER for a number of experimental and numerical studies. The general trend in experimental results7,29,30 is that the CP,max grows with ER . The simulation of Le 22 and current result fit into that trend, however the result by Barri et al. 3 shows a reduced pressure maximum with ER = 2. This effect could be due to lower Reh than in current simulation and experiments. Another comparison is presented in Fig. 14, where we plot the pressure coefficient at the bottom wall against the location scaled with the reattachment position x∗ = x−Xr Xr . Since the reference simulation and experiments were conducted using different expansion ratios, we use the scaling7 : CP∗ =
CP − CP,min , CP,BC − CP,min
W
to a new steady level (W represents the no-slip walls and τ is the viscous stress). To establish that the averaging time was sufficient for converged statistics, we looked
where CP,min is the minimum pressure coefficient and CP,BC = E2R 1 − E1R is the Borda-Carnot pressure coefficient. Our simulation collapses particularly well with
Simulation of turbulent flow over a backward-facing step at Re=9000
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FIG. 11. Spanwise averaged power spectrum density for velocity fluctuation u (a) and pressure fluctuation p0 (b) at different location in the flow field. Positions of the measurement points are depicted in the top panel. Point coordinates: #2 (x=0.1h, y=h), #5 (x=4h, y=h), #9 (x=8h, y=0.01h). Complete data for all measurement points is available in24
Le 22 , while the experimental data sets retain some of the ER dependence. The result by Barri et al. 3 (dot-dashed line) show significantly lower scaled CP than other results, even though it has the same ER as our simulation. Fig. 15 shows mean static pressure coefficient with a clear pressure drop zone originating at the step edge and spanning up to approximately x = 4.2h ≈ 12 Xr . The same pressure deficiency is visible in Fig. 16, which shows static pressure variations across the channel at four different locations in the outflow channel, using the static pressure Pw at the top wall for their respective locations as reference. The figure shows that the pressure deficit in
the recirculation zone is mainly in the mixing layer and there is a significant difference in the static pressures at the top and bottom wall throughout the outflow channel. In the recirculation zone (x = 0.5h, 4h) the difference is in favour of the top wall, while in the reattachment zone (x = 8h) the static pressure at the bottom wall is higher. Far downstream (x = 20h) the static pressure profile slowly returns towards a uniform distribution across the channel.
Simulation of turbulent flow over a backward-facing step at Re=9000
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6
PSD(u')
-4
1.5
0
(a)
×10
0.053 4 0.142 2 0
4
0.048
2 0
10
-1
10
St = f h / U
0
10
-1
St = f h / U
b
b
0
FIG. 12. Spanwise averaged power spectra of streamwise velocity fluctuation u (a) and pressure fluctuation p0 (b) for two points of the the additional simulation. The points coordinates are defined in figure 11
C P,max
1
current
0.4 Westphal Kim et al. (1980) et al. (1984) Le (1995)
0.2
0.8
Barri et al. (2010)
0.6 CP∗
Driver & Seegmiller (1985)
0 1
1.4
ER
1.8
2.2
0.4 0.2
FIG. 13. Maximum of static pressure coefficient as a function of expansion ratio in different with experiments and simulation experiments and simulations CP,max against the expansion ratio ER ;
2.
Velocity field
Fig. 17(a) plots the mean streamwise velocity in the recirculation and reattachment with the incoming flow slowly expanding towards the bottom wall, reattaching to it around x = 8.62h then regenerating further downstream into a fully developed channel flow. Immediately after the step, the interaction of the incoming flow, and the fluid trapped in the corner allows a recirculation bubble to form. The maximum of this reverse flow occurs between x = 2.0h and x = 6.0h with Umin ≈ −0.25Ub at x = 3.91h, y = 0.08h. There is no evidence for a recirculation bubble on the top wall, which is in agreement with earlier work22 . In addition to the primary recirculation bubble, streamlines show several additional eddies in the step corner and along the bottom wall (Fig. 17b, c). Based on the U = 0 isoline, the secondary corner eddy spans up to x = 0.99h in the streamwise direction and y = 0.8h vertically, which is in excellent agreement with previous
0 -1
0
1 (x-X r )/X r
2
FIG. 14. Pressure coefficient at the bottom wall from Driver and Seegmiller 29 (crosses); Kim, Kline, and Johnston 7 (circles); Westphal, Johnston, and Eaton 30 (diamonds); Le 22 (dashed line) and Barri et al. 3 (dot-dashed line). Current simulation represented by a solid line.
simulations1,3 . The centre of the secondary corner eddy is located at x = 0.328h, y = 0.243h. Fig. 17c shows a tertiary corner eddy, which resembles the prediction by Moffat 31 made for the low Reynolds number flow in the vicinity of the sharp corner. The theory predicted an infinite number of eddies decreasing in size and strength in the limit of Re → 0. Previous computations32 showed two corner eddies for Re = 1. However, experiments by Hall et al. 15 investigated the secondary vortex in the turbulent backward-facing step flow but did not reveal any tertiary eddies. In our simulation, the tertiary corner eddy size is 0.062h in horizontal and 0.11h in the vertical dimension. Its centre is located at x = 0.03h, y = 0.042h. This result is in good agreement with LMK1 , who reported the presence of a tertiary corner eddy of 0.042h
Simulation of turbulent flow over a backward-facing step at Re=9000
10
0.4 0.3 2 y/h
0.2 0.1 0
0 0
5
x/h
15
10
-0.1
FIG. 15. Mean static pressure coefficient contours. CP =
2
y/h
y/h
2
P −P0 1 ρU 2 b 2
1
0
1
0 0
0 0 (P - Pw)/(0.5 ρ U2b)
0
0
FIG. 16. Static pressure variation (P − Pw )/(0.5ρUh ) across the channel at four different positions (from left to right): x/h = 0.5, x/h = 4.0, x/h = 8.0 and x/h = 20
0
U/Ub0
0
FIG. 18. U velocity profiles at four different positions (left to right): x = 0.5h, x = 4h, x = 8h and x = 20h. The distance between vertical grid lines represents U/Ub =.
(a)
y/h
1
0 0
2
(b)
6
4 x/h
(c) 2
y/h
y/h
0.8 1
0.2
0
0 0
x/h
1
0
0.1 x/h
0.2
FIG. 17. U velocity field (colormap) and streamlines (black solid lines). The red solid line marks the U = 0 isoline. (a) The recirculation area. (b) Secondary recirculation bubble with an additional structure (eddy extension) between x/h = 1.0 and x/h = 1.5. (c) Tertiary corner bubble.
in size. Closer examination reveals an additional secondary structure at the tip of the secondary corner eddy. It has the same anti-clockwise direction as the main secondary eddy and spans 0.99h < x < 1.44h, with a center
at x = 1.237h, y = 0.025h (Fig. 17b). We refer to this structure as a secondary eddy extension. Consistent with these findings are PIV measurements by Hall et al. 15 , which indicate that an additional secondary structure might be present in the BFS flow. Their results show that at the tip of the secondary eddy a part of the primary recirculating flow turns just ahead of the secondary vortex and flows in the direction perpendicular to the cross-sectional plane. They argued that this is unlikely to be a result of PIV error and concluded, that this might indeed be a new flow structure. This structure coincides in space with the additional secondary vortex revealed by the present study. Fig. 18 shows U profiles at different x locations. Initially the fully developed turbulent flow expands freely into the expanded channel at x = 0.5h. Area of reversed flow visible at x = 4h, but disappears at x = 8h, despite the wall shear stress (Fig. 7, Table I) indicating that the mean reattachment position is at Xr = 8.62. As the profiles move further downstream, they slowly return to those for equilibrium channel flow. Even at x = 20h the fully developed turbulent channel flow profile has not been reached, in agreement with Le, Moin, and Kim 1 . Experiments6 also report that even at long distances downstream (50h) the velocity profile is still not fully recovered. Fig. 19 plots profiles of the mean vertical velocity V . Shortly downstream of the step (x = 0.5h) there is a a
Simulation of turbulent flow over a backward-facing step at Re=9000 2
11
(a)
1 y/h
y/h
2
1
0 0.05
0
V/U b0
0.05
0
0.05
0
FIG. 19. V velocity profiles at four different positions(left to right): x/h = 0.5, x/h = 4.0, x/h = 8.0 and x/h = 20
(b)
strong V gradient in the mixing zone, with a strong upward motion in the bottom part of the channel. Between x = 4h and x = 8h there is a clear downward movement corresponding to the inlet jet reattaching to the bottom wall. The downward tendency, although minimal, is still present as far as x = 20h downstream of the step.
y/h
0
0
0.03
0
0.03 0 √ uu/Ub
0.03
0
0.03
0
0
0.015
0
0.015 0 0.015 √ vv/Ub
0
0.015
0
0
0.1
0
0.1 0 √ ww/Ub
0.1
0
0.1
0
0
0.01
0
0.01 0 - uv/Ub2
0.01
0
0.01
0
2
1
0
(c) 2
Turbulence intensity and Reynolds shear stress
y/h
3.
Turbulent intensity and Reynolds shear stress profiles at four x locations along the channel are shown in Fig. 20. All quantities show √ a sharp spike in the mixing layer at x = 0.5h, with w0 w0 significantly larger than other components. The spike widens gradually downstream, but - similarly to velocity profiles in Fig. 18 - the turbulent channel flow profiles are not fully recovered at x = 20h.
0
(d)
y/h
2
Even though with sufficiently long averaging times we expect the flow to be homogeneous in the spanwise direction, it is worthwhile to look at time-averaged spanwise structures that remain after the averaging used in the current simulation. In Fig. 8 we see that the reattachment line shows a wavy structure in the spanwise, with clearly visible 3-4 lobes. To investigate this further, we plot in Fig. 21 a y − z cross-section of time-averaged vertical velocity taken at x = 6h, shortly upstream from the mean reattachment position. The solid black line marks the location of the U = 0 isosurface, indicating the boundary of the recirculation bubble. The average downward tendency areas overlap with the three to four downward lobes in the recirculation bubble. Those minima correspond, in turn, to the locations of the spanwise minima of the reattachment line in Fig. 8. Fig. 22 shows the time averaged vortex criterion33 λ2 plotted in the same setting as Fig. 21. This criterion identifies the low pressure zones typically associated with strong vortices. Although less clear than vertical velocity, three to four positive-negative pairs of λ2 can be found near z/h =3.6, 4.8, and less distinctively around z/h =1 and 6. This would indicate four
1
0
Persistent streamwise vortices
√ √ FIG. 20. Turbulence intensity profiles u0 u0 /Ub , v 0 v 0 /Ub , √ w0 w0 /Ub and Reynolds stress profiles u0 v 0 /Ub at four locations (left to right): x = 0.5h, x = 4h, x = 8h and x = 20h
2
0.06 0.02
y/h
4.
1
-0.02
1
-0.06 -0.12 0 0
2
z/h
4
6
FIG. 21. Spanwise structure of the time-averaged vertical velocity at x = 6h for the Lz = 2πh Re = 9000 calculation. Four subsiding (blue) structures can be identified across the y/h = 0.6 line at z/h=1.6, 3.6, 4.8 (weakly) and crossing the periodic boundary at z/h = 0 = 2π. The upwelling (yellow) zones are less distinct, but four zones between the subsiding structures can be identified at z/h =1, 2.6, 4 and 5.3.
Simulation of turbulent flow over a backward-facing step at Re=9000 2 0.02
y/h
1.5
0.01 0
1 -0.01 0.5
-0.02 -0.03
0 0
1
2
3 z/h
4
5
6
FIG. 22. Spanwise structure of the time-averaged vortex core criterion λ2 at x = 6h for the Lz = 2πh Re = 9000 calculation. (a)
(b)
FIG. 23. Spanwise structure of the time-averaged vortex core criterion λ2 at x = 6.0h: a) The Lz = 0.75π simulation. b) The Lz = 1.25π simulation. Bold solid line marks U = 0. Two streamwise vortices can be identified for each, but only for Lz = 1.25π is the spacing roughly the same as for the Lz = 2πh calculation.
streamwise vortices present in the flow, which are persistent enough to not be filtered out by the time-averaging. These structures cannot be clearly seen when individual times are plotted which raises these questions: Is the observed spanwise structure a physical phenomenon, or is it an artifact of the imposed spanwise periodicity and the time-averaging? To determine how the periodicity of the domain affects the streamwise structures, we have run two additional simulations with reduced spanwise dimension Lz = 0.75πh and Lz = 1.25πh. To reduce the computational expense, their initial conditions were generated from the original Lz = 2πh, Nz = 128 simulation by keeping only the first 24 and 40 Fourier modes respectively, adding some random noise, and running until T = 120h/Ub . Structures can be inferred by comparing the variations in the U = 0 line, which indicate recirculation lobes, with the λ2 criterion. Using this comparison, Fig. 23a has two structures over a much shorter spanwise spacing that was used by the Fig. 22 calculations. In contrast, even though Fig. 23b also has two dominant
12
structures, this is with nearly twice the spanwise domain of 23(a) and roughly the same spanwise spacing of the structures as in Fig. 22. This allows us to conclude that the spacings in Fig. 22 for the primary Lz = 2π case are physical. The time averaged spanwise energy spectra taken in the mixing layer in Fig. 3c show clear peaks of energy in w0 and u0 for wavenumbers kz = 2 and kz = 3, which correspond to two or three wavelengths in the spanwise direction. The energy spectrum of v 0 lacks a peak but shows a plateau for kz ≤ 3 and turbulent-like powerlaw decay for kz > 3. The average over all y of the v 0 spectra at x = 4h has a similar plateau, which indicates that large excursions in V at x = 4h, potentially analogous to those for U and W , have been suppressed by the streamwise vortical structures. Together, the contour plots in Fig. 21, 22, 23, plus the V spectrum in Fig. 3d, support a view that the time-averaging is acting like a coarse-grained filter that removes the small-scale turbulent fluctuations, thereby allowing us to see the persistent large-scale structures. The presence of persistent vortices does not mean, however, that they would remain in the same positions. We believe that over very long averaging times the V = 0 profiles in Fig. 21 and U = 0 in Fig. 8 would be homogeneous in the spanwise direction. Barri et al. 3 does not report such structures while using an order of magnitude larger averaging time than is employed in this paper. We note that in the stability analysis of Barkley, Gomes, and Henderson 34 , the leading instability modes took the form of steady elongated three-dimensional rolls that were largely confined to the separation bubble attached to the step, and that these general features appear similar to the persistent structures we have observed here. Barkley, Gomes, and Henderson 34 suggested that the underlying mechanism for the steady instability mode is centrifugal, but confined within the separation bubble and so not of the Taylor–G¨ortler type. However, the spanwise wavelengths here are of the order two step heights, compared to an onset wavelength of order seven step heights in the steady laminar flow34 . Without further analysis, it is difficult to be categorical about the linkage between the two studies, or physical mechanism in the present case.
IV.
CONCLUSIONS
Numerical simulation of turbulent flow over a backward-facing step was performed for Reynolds number of Reh = 9000, with enough spatial and temporal resolution to allow the first numerical investigation of the interaction of the streamwise and spanwise vortices with the reattachment position. Both the average position of the reattachment of the recirculation eddy and its frequency are consistent with experiments. A crucial part of achieving this was providing a turbulent flow into the domain. To accomplish this, the
Simulation of turbulent flow over a backward-facing step at Re=9000 inflow channel was extended to x = −12h, and the velocity boundary condition was regenerated from a plane at x = −4h, forming a periodic channel with length 8h. To keep the mass flow constant, Green’s function corrections were applied to the periodic flow, yielding accurate flow rate control with a little extra computational expense. With this turbulent inflow, the calculations were able to reproduce the streamwise velocity profiles and turbulence intensity profiles reported in literature25,26 . The following properties agree with experiments and previous computations, validating this approach: The mean reattachment length Xr = 8.62h for ER = 2.0 and Reh = 9000 matches the values from Armaly et al. 9 for a two-dimensional turbulent flow over a BFS with ER = 2.0. Furthermore, taking into account the differences in Reh and ER and scaling the abscissas by Xr , the streamwise profile of the coefficient of friction at the bottom wall agree with previous experimental and computational results, as does the position of the maximum negative skin friction at 0.62Xr and the dependence of the value of maximum negative peak on Reh 1,10,11,13 (see Fig. 6 and Tab. I). For pressure statistics, the coefficient of pressure at the bottom wall obeys the scaling of Kim, Kline, and Johnston 7 as does the position of maximum CP with respect to Xr , confirming the strong dependence upon ER found in previous work for lower expansion ratios7,22,29,30 (see Fig. 13). What these calculations can now show is that in addition to the main recirculation bubble, the time and spanwise averaged velocity field provides evidence for the secondary and tertiary corner eddies with this additional feature: paired with the secondary eddy, an additional vortical structure appears that is located at its downstream tip and rotating in the same direction. Weak evidence for such a structure has appeared in experiments15 , but the PIV measurements were not conclusive. Besides confirming the experimental observation, Fig. 21 identifies some of the spanwise structure within the secondary eddy using time averages of the velocity field V and vortex core criterion λ2 33 . Three to four clusters of positive and negative pairs are noted, along with a similar structure appearing as lobes along the reattachment line in Fig. 8. Figure 23 indicates that the spacing of these lobes is physical, based on the changes in spacing between the lower Lz calculations. Perhaps the most complex dynamics identified by these simulations are the coherent oscillations and other quasiperiodic behavior associated with reattachment. This includes oscillations in the reattachment position, a sawtooth shape for the wall shear stress associated with this reattachment and reattachment flapping. Comparisons between the frequencies reported here and experimental frequencies are in qualitative agreement, with the turbulent inflow being crucial in achieving this agreement as a similar match was not found for simulations with laminar inflows28 . This shows that when attempting to reproduce experimental results, one needs to do more than just match the domain, essential boundaries, and
13
Reynolds numbers. The nature of the external forcing or inflow must also be taken into account. 1 H.
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