Spectral Element Simulation of Flow Around a Surface-Mounted Square-Section Cylinder Johan Malm1 , Philipp Schlatter1 , Dan S. Henningson1 Lars-Uve Schrader2 , Catherine Mavriplis2 1
Linné FLOW Centre, Swedish e-Science Research Centre, KTH Mechanics, SE-100 44 Stockholm, Sweden 2 Department of Mechanical Engineering, University of Ottawa, Ottawa, Canada K1N 6N5 Email:
[email protected]
A BSTRACT A three-dimensional direct numerical simulation (DNS) of the flow around a square-section cylinder mounted vertically on a flat plate is presented. The simulation data will be part of the CFDSC 2012 Challenge. The open-source spectral element code Nek5000 is used to solve the incompressible Navier– Stokes equations. A “precursor” time-dependent inflow condition of high quality in the form of a turbulent boundary layer is generated by a pseudo-spectral Fourier-Chebyshev code, which considerably reduces the computational domain and thereby the computational cost of the main simulation. The flow behind the cylinder is massively separated and characterized by a von Kármán vortex street. In our simulation the Strouhal number of the vortex shedding is St ≡ f d/U = 0.11, which is in close agreement with the value of St = 0.1 ± 0.03 reported in the windtunnel experiment. The same period of the vortex shedding is present from the top of the cylinder all the way down to the turbulent boundary layer. The three-dimensional nature of the vortex shedding is manifested through an inclined wake of approximately 45 degrees due to the friction at the lower wall, which results in a phase difference of approximately 90 degrees between the vortex shedding cycle at the top of the cylinder and that inside the boundary layer. Preliminary mean flow results in selected planes show good agreement with the available experimental data already after four shedding periods.
1
F LOW C ONFIGURATION
The CFDSC 2012 Challenge deals with the numerical simulation of the flow around a square-section cylinder mounted vertically on a flat plate. A windtunnel ex-
periment on the same geometry placed in a zero pressure gradient (ZPG) turbulent boundary layer (TBL) serves as a benchmark. The reported value for the freestream turbulence is Tu = 0.8 %. The horizontal coordinates are denoted by x (streamwise) and y (spanwise), and the vertical coordinate (parallel to the cylinder axis) is z. The cylinder features a quadratic cross section of d 2 and a height of 4d. The Reynolds number is Re = Ud/ν = 11000, where U denotes the freestream velocity and ν is the kinematic fluid viscosity. In the experiment the boundary-layer thickness at the middle of the cylinder is reported to be δ99 = 0.72d, which corresponds to a Reynolds number based on momentum thickness θ and freestream velocity U of Reθ ≈ 1000. Keeping in mind that one of the largest direct boundary-layer simulations conducted so far reached up to Reθ = 2500 [16], this set-up poses a veritable challenge for any eddy-resolving numerical technique. At this Reynolds number, the flow is dominated by a von Kármán vortex street, in which vortices are shed at a well-defined Strouhal number in the turbulent wake. A wide spectrum of length scales — ranging from small turbulent structures in the shear layers to large vortices in the cylinder wake — is expected, which poses further challenges for the numerical method. We present here a direct numerical simulation (DNS) of this flow, aiming at resolving all spatial and temporal length scales.
2
N UMERICAL M ETHOD AND S IMULATION S ET-U P
Since compressible effects are assumed to be small, we are solving the incompressible Navier–Stokes equations using the high-order spectral element code Nek5000, developed and maintained by Fischer et
presented in this paper, approximately 0.8 million core hours have been used.
z " x ! Figure 1: Simulation set-up showing the current domain (solid). Data from the fully spectral precursor simulation (dashed domain) is provided as a timedependent inflow condition at plane A. al. [5]. It is based on the spectral element method (SEM) [13], which combines the high accuracy of global spectral methods with the geometrical flexibility of finite element methods (FEM). As for FEM, the computational domain, sketched in Fig. 1 in a spanwise-constant plane, is decomposed into K elements, where K = 145, 744 in the present simulation. In each element the governing equations are written in the weak form and discretized by a Galerkin procedure, where test and trial functions are sought in different polynomial spaces PN and PN−2 of maximum order N and N − 2 for velocity and pressure, respectively [9]. This results in a staggered pressure grid with regard to the velocity grid, obviating the possibility of spurious pressure modes. The solution of the velocity u = (u, v, w) is represented by tensor products of Legendre polynomial Lagrangian interpolants hNi (x) of maximum order N. In a single element Ωe , e = 1, ..., K, the approximation reads: N
u(xe (r, s,t))|Ωe = ∑
N
N
∑ ∑ ueijk hNi (r)hNj (s)hNk (t)
(1)
i=0 j=0 k=0
where xe is the coordinate mapping from the referˆ to the local element Ωe and ue is ence element Ω i jk the nodal basis coefficient. For the present simulation, we choose N = 11 for the velocity grid and N = 9 for the pressure grid, resulting in approximately 250 million points. The simulation code Nek5000 employed here uses an efficient MPI-based parallelization that has proven adequate for direct numerical simulation of flow configurations with millions of degrees of freedom [11]. The simulation is running on the AMD cluster ‘Ekman’ using 2048 cores. Excellent speedup on this architecture has been observed [10]. For the result
In the experiment the leading edge of the zero pressure gradient boundary layer is reported to be 16d upstream of the cylinder. In order to reduce computational cost, the numerical domain for the present DNS does not contain the leading edge itself but starts 8d downstream of the leading edge, as indicated by the solid rectangle in Fig. 1 with sides A, B, C and D. This is accomplished by the use of the more efficient (but less flexible) Fourier-Chebyshev spectral code SIMSON [3], which generates time-dependent Dirichlet conditions ahead of the main simulation. The domain pertaining to this code is indicated by the dashed region in Fig. 1: It starts with a laminar Blasius profile at x/d = −28 (Reθ = 180), spans a downstream distance of 33d and ends around Reθ = 1000. (Note that the cylinder is not part of this domain.) The domain of the precursor simulation is discretized using 1024 × 201 × 768 grid points in the streamwise, wallnormal and spanwise directions, respectively. Transition to turbulence is triggered by means of a tripforcing technique approximately at the location of the arrow (Tr). This technique was successively employed in e.g. [16] for the simulation of spatially developing turbulent boundary layers. The auxiliary simulation is designed to produce the correct boundary-layer thickness of δ99 = 0.72d at x/d = 0, as measured in the experiment. The dashed line in Fig. 2(a) computed from the spectral data certifies that if we start the auxiliary box at x/d = −28, then Reδ99 /7.2 = 1100 at x/d = 0. Taking δ99 = 0.72d at the position of the cylinder, this gives us a Red = 11000, which is the Reynolds number reported from the experiment. The velocities in the plane x/d = −8 (or Reθ = 790 according to the solid line in Fig. 2a) are stored with interval t ∗ = 2U/d. The turbulent mean velocity profile of the inflow data at this streamwise location is shown in Fig. 2(b), where excellent agreement with the law of the wall can be inferred. The slope of the log law is chosen to be 1/κ = 1/0.41 and the intersection with the y-axis is B = 5.2 [15]. The Reynolds stresses are shown in Fig. 2(c). The inflow data is interpolated in space to match the current SEM grid, exemplified by a series of three fields in Fig. 3. During the main simulation, Lagrangian interpolation using a third order polynomial is performed at every time step between two given planes of data and then applied at the inflow boundary A in Fig. 1. This technique establishes an improvement to the so-called recycling techniques (see e.g. [7]) in that the inflow data is not recycled here:
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Figure 3: Time-dependent inflow condition for u at three different times of the zero pressure gradient turbulent boundary layer 8d upstream of the cylinder generated by the Fourier-Chebyshev spectral code. Values range from 0 (blue) to 1 (red). Lagrangian interpolation in time between the given planes of data is performed in every time step using a third order polynomial.
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Figure 2: (a) Development of the ZPG TBL computed by the Fourier-Chebyshev spectral code SIMSON [3], where the solid line shows Reθ (x/d) and the dashed Reδ99 (x/d)/7.2. The dotted parts of the lines indicate the development upstream of the SEM domain. (b) Turbulent mean velocity profile compared to the log ) and (c) normal stresses (u+ , w+ law ( rms rms + & & + , vrms ) and shear stress ({u v } ) from the precursor simulation normalized by friction velocity and plotted as a function of wall-normal distance normalized by the viscous length scale at Reθ = 790 where the inflow plane to the main simulation is located.
The auxiliary simulation provides a sufficiently long time history of the inflow so that unphysical behavior due to an artificial recycling frequency is avoided. No-slip conditions are applied at the wall (C in Fig. 1)
and no-stress conditions are applied at the top boundary (B) and at the outflow (D). These boundaries are located 12d above the lower wall and 16d downstream of the obstacle, respectively, as indicated in Fig. 1. A thin sponge region (length 2d) upstream of the outlet at D creates a disturbance free outflow, ensuring numerical stability (see e.g. Ref. [4]). In this region the flow is forced to the spanwise mean flow given at the present time, which guarantees the correct mass flux. Finally, periodic boundary conditions are applied in the spanwise direction (y). The spanwise width of the computational domain is 16d. Due to lack of information regarding the turbulent length scale in the freestream, the freestream is in our set-up disturbance free, i.e. Tu = 0 %.
3 3.1
R ESULTS
Instantaneous Flow
The quality of the time-dependent inflow condition can be assessed via the mean development of the boundary layer in the streamwise direction far from the obstacle in the spanwise direction. This is deferred to the next section where general mean flow properties are discussed. Looking at the instantaneous results, we note the occurrence of turbulent near-wall streaks, emerging from the inlet boundary. They constitute an essential part of any correctly represented zero pressure gradient turbulent boundary layer. In Fig. 4 they are visualized by means of isosurfaces of streamwise velocity
# ! λ/2
Figure 4: Isosurfaces of instantaneous streamwise velocity u = 0.4U.
u = 0.4U and in Fig. 5 in a wall-parallel plane colored by streamwise velocity at z+ ≈ 15. They appear straight and undisturbed approximately up until 2.5d ahead of the obstacle (x/d = −3), after which they are slightly bent and deflected due to the presence of the cylinder (Figs. 4 and 5). At a spanwise distance of approximately 5.5d from the cylinder (y/d = ±6) the streaks are straight and the boundary layer is relatively undisturbed. Upstream of the cylinder the flow stagnates, which gives rise to a thickening of the incoming boundary layer, visible in Fig. 4. Above a height of approximately 0.8d, the stagnating flow on the front side of the cylinder is laminar. Transition to turbulence takes place in the thin shear layers on the sides and the top of the cylinder that bound the separation bubbles on these surfaces. The point of separation on the cylinder is fixed to the sharp upstream edges (two vertical and one horizontal), thereby falling into the category of so-called ‘geometry-induced’ separation. The flow behind the cylinder is massively separated with large areas of negative flow, as shown in Fig. 6, where the wall-parallel plane in Fig. 5 is combined with the symmetry plane at y/d = 0 and one crossflow plane at x/d = −8. On both sides of the cylinder (in the flatplate boundary layer) the flow is accelerated, and the streamwise streaks are suppressed. Since it has been observed that in favorable pressure gradient boundary layers the near-wall streaks are elongated but not necessarily weaker [14], the cause for this weakening may have another origins — possibly the increased streamwise vorticity in this region. As expected for this type of cylinder flows, a von Kármán vortex street is one of the dominating features. At this Reynolds number, the wake is fully turbulent and
Figure 5: A wall-parallel plane at z+ ≈ 15 showing streamwise velocity ranging from −0.3 (blue) to 1.2 (red).
Figure 6: A wall-parallel plane at z+ ≈ 15, a crossflow plane at x/d = −8 and the symmetry plane at y/d = 0 showing streamwise velocity ranging from −0.3 (blue) to 1.2 (red). Note that the cylinder is transparent.
has a relatively sharp interface to the irrotational surrounding flow, as can be seen in Fig. 7 where streamwise velocity in a wall-parallel plane at z/d = 2 is shown. From Fig. 7 we also note that the streamwise wavelength of the vortex street is λ ≈ 10d. It is interesting to observe that the von Kármán vortex street has a large impact on the flow close to the wall. This can be seen by comparing Fig. 5 (near-wall plane) with Fig. 7 (plane outside the boundary layer), which pertain to the same instant of time: The wake structures in the boundary layer (Fig. 5) feature a similar (slightly shorter) wavelength compared to the vortex street in the outer flow (Fig. 7). Moreover, we notice a phase shift between the flow structures in these two planes. This phenomenon is confirmed and quantified by signals obtained from a set of time probes,
1.05 1.04
u
1.03 1.02 1.01 *
3T
1
which monitor the flow velocity vector in the domain. First, the frequency of the vortex shedding is established by means of the streamwise velocity component u = u(x p , y p , z p ,t). The unsteadiness induced by the vortex shedding is also sensed in the freestream around the cylinder, where the signal is free from turbulence and hence easier to evaluate. Almost five shedding periods are visible in Fig. 8 corresponding to a probe location of (x p , y p , z p ) = (7d, 4.2d, 3d). An estimate of the length of one period is given by the mean over three periods (indicated in Fig. 8), such that 3T ∗ ≡ tU/d ≈ 27.27 → T ∗ = 9.09. This corresponds to a non-dimensional frequency St ≡ f d/U = 1/T ∗ ≈ 0.11, which is close to the Strouhal number of St = 0.1 ± 0.03 reported in the experiment. Assuming a convection velocity of uc ≈ 1, this frequency will give rise to the spatial wavelength observed in Fig. 7 as λ = uc / f ≈ 1/0.11 ≈ 9d. It also explains the slightly shorter wavelength observed in Fig. 5, since there uc < 1 and hence λ < 9d (assuming a fixed frequency f , which is a valid assumption, as shown below). A Strouhal number of 0.1 is a factor two lower than the well-documented value of 0.2 for circular cylinders. This has been observed in other studies of square cylinders, e.g. in Refs. [17, 8, 6]. The reason for this is most likely linked to the differences in the separation process: Whereas for a circular cylinder the separation occurs from a smooth surface, which makes the point of separation vary in the azimuthal direction, it is fixed to the upstream corners of the square cylinder — at least at the present Re. If we now, for a fixed streamwise and spanwise location, include probes with a varying wall-normal coordinate, we will get a collection of curves seen in the panel to the left of Fig. 9. Only one period is shown
70
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tU/d
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Figure 8: Time probe history of streamwise velocity u = u(x p , y p , z p ,t), where (x p , y p , z p ) = (7d, 4.2d, 3d). 3 non-dimensional periods of 3T ∗ ≈ 27.27 → T ∗ = 9.09, which gives a Strouhal number of St = 0.11 is indicated. and the signals are obtained at z/d = 1, 2, 3, 4.2. The arrow indicates that the peaks owing to the various signals are slightly shifted as one progresses away from the wall, also noted by [2]. The period of the signals is however constant, indicating the presence of a ‘global’ mode. A constant frequency is in agreement with Ref. [1], but contradicts Ref. [12], where a varying shedding frequency along the cylinder height was suggested. In the panel to the right of Fig. 9, the signal at z/d = 0.027 (z+ ≈ 15) is compared to that at z/d = 4.2. The signals are approximately 90 degrees out of phase, which was also concluded from Figs. 5 and 7. This behavior manifests the three1.15 1 1.1 0.8 1.05
u
Figure 7: A wall-parallel plane at z/d = 2 showing streamwise velocity ranging from −0.3 (blue) to 1.2 (red).
60
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" ! λ/2
0.99 50
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Figure 9: Time probe history of streamwise velocity u = u(x p , y p , z p n ,t), where x p = 7d, y p = 3d and in ), z p 2 = 3d ( ), z p 3 = panel (a) z p 1 = 4.2d ( 2d ( ), z p 4 = 1d ( ). In panel (b) z p 1 = 4.2d ( ), and z p 2 = 0.027d ( ). dimensionality of the von Kármán vortex street. In-
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Figure 10: A streamwise-constant plane at x/d = 7 showing a snapshot of the inclined wake structure by means streamwise velocity ranging from −0.3 (blue) to 1.2 (red). The solid line indicates an angle with the lower wall of 45 degrees.
0.5
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deed, for an infinitely long cylinder, the periodic vortex shedding would be in phase at all positions along the height of the cylinder. The presence of the wall in the present case introduces friction, which retards the flow near the wall with respect to the outer flow. The result is an inclination of the instantaneous wake structure by approximately 45 degrees. A snapshot of the wake is shown in Fig. 10 at one of the outer positions of the shedding cycle. The solid line indicates an angle with the lower wall of 45 degrees.
3.2
Mean Flow
To further assess the quality of the inflow condition and ensure that the correct streamwise development of the boundary layer is obtained, the δ99 thickness is measured in the (assumed) undisturbed part of the boundary layer (y/d = 8) at the streamwise location (x/d = 0) of the cylinder. The thickness of the boundary layer impacts the actual Reynolds number of the simulation in the following way: Resim = exp exp = 11000, Reexp × (U/1) × (δsim 99 /δ99 ), where Re exp δ99 = 0.72d are the reported values from the experiment. In the present simulation, we get δ99 = 0.69d and hence Resim = 10543, which is very close to the nominal value of Re = 11000 obtained in the fully spectral auxiliary simulation at the same streamwise position. The discrepancies can be explained by residual effects of the cylinder. Preliminary mean flow results in the wake of the cylinder are given in Fig. 11(a), with a comparison to the experimental data in Fig. 11(b). The data are shown in the symmetry plane y/d = 0. The data from our numerical simulation was averaged over a time t ≈ 39d/U, i.e. roughly four shedding periods. Despite this relatively short averaging time the range of the data is in close agreement with the experimental coun-
0
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Figure 11: Streamwise mean flow component in the symmetry plane y/d = 0 in the wake of the cylinder given by (a) our numerical calculation and (b) the experiment.
terpart and all essential features are present. For example, the narrowly spaced contours from approximately (x/d, z/d) = (2, 4) to (x/d, z/d) = (4, 2) share the same angle of 45 degrees in both data sets. After this point, they approach the wall vertically for a short distance, but then attach to the wall at an angle due to faster moving flow in the boundary layer. The minimum velocity in the wake flow is located slightly closer to the wall in our data and can further be seen to be somewhat lower than the experimental counterpart. This minimum is expected to approach the experimental data after a longer averaging period, since extreme velocities usually smooth out during this process. The small region with positive flow right behind the cylinder in the flat-plate boundary layer is visible both in the numerical and in the experimental data. Also in a streamwise-constant plane at x/d = 5 shown in Fig. 12, the comparison is satisfactory notwithstanding the limited averaging. (Note that the data shown here and in Fig. 13 was averaged in space with respect to the symmetry plane.) For instance, the peaked contour in the symmetry line at z/d = 3.5 is present. In the experimental data velocities just below the freestream velocity (∼ 0.99U) are observed around the actual wake, which is not the case in our data. This is one of the differences that will most likely not disappear as the averaging of the numerical data is continued. One potential reason for this discrepancy could be the effect of the freestream turbulence (0.8 % at the position of the
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Figure 12: Streamwise mean flow component in a streamwise-constant plane at x/d = 5 in (a) our numerical calculation and (b) the experiment.
−3 −4
cylinder), which is present in the experiment but not taken into account in our simulation. Finally, we compare the spreading of the wake in a wall-parallel plane shown in Fig. 13. We note that the spreading rates are in close agreement, save that our numerical data, as noted above, show a more undisturbed freestream than the experimental data does. To summarize, the mean flow results presented here are in good agreement with the benchmark results provided by the windtunnel experiment. However, the final data is expected to get much smoother with increased averaging time.
4
C ONCLUSIONS
A direct numerical simulation of the flow around a surface-mounted square-section cylinder with a highorder spectral element code has been performed. The data will be part of the CFDSC 2012 Challenge, where it will be presented and discussed. Inflow conditions are generated by a spectral Fourier-Chebyshev code, which reduces the computational cost of the main simulation and provide a highly accurate representation of the incoming zero pressure gradient turbulent boundary layer. The flow behind the cylinder is turbulent and mas-
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Figure 13: Streamwise mean flow component in a streamwise-constant plane at z/d = 1 in (a) our numerical calculation and (b) the experiment.
sively separated, but nevertheless a distinct vortex shedding frequency of St = 0.11 is found. This value is in good agreement with the experimental findings, where St = 0.1 ± 0.03 is reported. Due to the interaction with the incoming boundary layer and the horizontal wall itself, the two-dimensionality of the vortex shedding is lost and the shedding at various wallnormal coordinates are slightly out of phase. The maximum phase difference is 90 degrees between the vortex shedding closest to the wall and one at the top of the cylinder. Despite an averaging time of only four shedding periods, the mean flow data is in good agreement with the experimental findings. A longer averaging period is needed and is currently in preparation in order to get a fully converged mean flow for the final presentation. Moreover, the Reynolds stresses will be evaluated and compared to the experimental data. Finally, proper orthogonal decomposition (POD) will be used to analyze the coherent structures in the wake.
ACKNOWLEDGEMENTS The authors gratefully acknowledge funding by VR (The Swedish Research Council) and NSERC (Natural Sciences and Engineering Research Council of Canada). Computer time was provided by SNIC (Swedish National Infrastructure for Computing) with a generous grant by the Knut and Alice Wallenberg (KAW) Foundation. The simulations were run at the Centre for Parallel Computers (PDC) at the Royal Institute of Technology (KTH).
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