UNSTEADY SEPARATED FLUID FLOWS AROUND A SPHERE
IN THE WIDE RANGE OF THE REYNOLDS AND FROUDE
NUMBERS
Valentin A. Gushchin and Paul V. Matyushin
Institute for Computer Aided Design of the Russian Academy of Sciences (ICAD RAS), 19/18, 2nd Brestskaya str., Moscow 123056, Russia
e-mail: gushchin@icad.org.ru, pmatyushin@mail.ru
Key words: Stratification, Formation Mechanisms of Vortices, Incompressible Fluid,
Numerical Simulation, Sphere Wake, Visualization
Abstract. The homogeneous (1 ≤ Re ≤ 5·106) and stratified (0.004 ≤ Fr ≤ 1, 10 ≤ Re ≤ 1000) viscous incompressible fluid flows around a sphere have been investigated by means of the direct numerical simulation (DNS) and the visualization of the vortex structures in the wake (where Reynolds number Re = U·d/ν, Froude number Fr = U/(N·d), d is the sphere diameter).
In case of the homogeneous fluid at 200 < Re ≤ 1000 the detailed formation mechanisms of
vortices (FMV) in the sphere wake have been described for the different unsteady periodical flow regimes (270 < Re ≤ 290, 290 < Re ≤ 320, 320 < Re ≤ 400, 400 < Re < 700 and
Re > 700) and the six basic FMV have been selected; at 290 < Re ≤ 320 a new flow regime
has been discovered; at 5·104 ≤ Re ≤ 5·106 the preliminary results are presented. The
following six basic FMV work during the period in the different parts of the flow: 1) in the recirculation zone a new vortex ring (or semi-ring) is generated near the sphere surface due to the Kelvin-Helmholtz instability (1k), the two vortex filaments connected with a new vortex ring are formed (1f), the main vortex ring is displaced closer to the sphere surface (1d); 2) the side parts of the vortex envelope (surrounding the recirculation zone) are stretched downstream (2s), the top or bottom edge of the vortex envelope is rolling up cylindrically and detached (2t/b); 3) in the outer flow the head of the vortex loop (facing upwards (t) or downwards (b)) is generated (3t/b). In case of the stratified fluid the four different flow regimes have been simulated (0.004 ≤ Fr ≤ 1, 10 ≤ Re ≤ 1000); for the first time the complex 3D vortex structures of these flows have been shown; the high gradient sheets of density have been observed near the poles of the resting sphere and the moving sphere (Fr ≤ 0.02). For DNS the explicit numerical method SMIF-MERANGE (second-order accuracy in space, minimum scheme viscosity and dispersion, monotonous) has been used. For the visualization of the vortex structures in the sphere wake the isosurfaces of the imaginary part of the
complex-conjugate eigen-values σ1,2 of the velocity gradient tensor have been used.
1 INTRODUCTION
sphere) are investigated by means of the direct numerical simulation (DNS).
Detailed experimental investigations of the vortex structures in the uniform homogeneous fluid flows around a sphere have been initiated in 1 and prolongated in 2,3 and other papers. As a result the following classification of the flow regimes has been obtained: 1) Re ≤ 200 – a steady axisymmetrical; 2) 200 < Re ≤ 270 – a steady double-thread; 3) 270 < Re < 300 - a double-thread with waves; 4) 300 < Re < 420 – a procession of the vortex loops (facing upwards) (Fig. 1); 5) 420 < Re < 800 – a procession of the vortex loops with the rotation of the shear layer; 6) 800 < Re < 3.7·105 - a procession of the vortex loops with the shear layer instability; 7) Re > 3.7·105 - the turbulent boundary layer. Detailed numerical investigations of the vortex structures in the uniform homogeneous fluid flows around a sphere have been initiated in 4 and prolongated in 5-10 and other papers. In spite of these papers the detailed FMV in the sphere wake are still unclear. At the present paper the detailed FMV in the sphere wake are described for 200 ≤ Re ≤ 1000; at 290 < Re ≤ 320 a new flow regime has been discovered; at 5·104 ≤ Re ≤ 5·106 the preliminary results are presented.
Detailed experimental investigations of the stratified fluid flows around a sphere have been carried out in 11 (0.005 ≤ Fr ≤ 20, 5 ≤ Re ≤ 10000) and other papers. The numerical studies of these flows are very rare. Therefore at the present paper the DNS of the stratified fluid flows around a sphere has been carried out (0.004 ≤ Fr ≤ 1, 10 ≤ Re ≤ 1000) in order to better understand the complex 3D vortex structures of these flows.
a)
b)
Figure 1: A procession of the vortex loops in the sphere wake, Re = 280, isosurfaces Im(σ1,2) = 0.02 during the
period: a) t = 1496.4, b) t = 1504.4.
2 NUMERICAL METHOD
The density of the homogeneous fluid ρ(x, y, z) = ρ0 where x, y, z are the Cartesian coordinates; z, x, y are streamwise, lift and lateral directions (Fig. 1); x, y, z have been non-dimensionalized by d/2 where d is the sphere diameter. The density of the linearly stratified fluid ρ(x, y, z) = ρ0(1 - x/(2C) + S) where C = Λ/d is the scale ratio, Λ is the buoyancy scale, which is related to the buoyancy frequency N and period Tb (N = 2π/Tb, N2 = g/Λ); g is scalar of the gravitational acceleration; S is a perturbation of salinity.
Navier-Stokes equations in Boussinesq approximation (1) – (2) and the diffusion equation for stratified component (salt) (3) with four dimensionless parameters: Fr = U/(N·d), Re = U·d/ν,
C, Sc = ν/κ = 709.22 where U is the scalar of the free stream velocity, ν is the kinematic
viscosity, κ is the salt diffusion coefficient.
g S Fr C Re p t g v v v v 2 2 2 -) ( ⋅∇ = ∇ + ∆ + + ∂ ∂ (1) 0 = ⋅ ∇ v (2) C S Re Sc S t S x 2 v 2 ) ( ∆ + ⋅ = ∇ ⋅ + ∂ ∂ v (3)
In (1) – (3) v = (vx, vy, vz) is the velocity vector (non-dimensionalized by U), p is a perturbation of pressure (non-dimensionalized by ρ0U2).
The homogeneous incompressible viscous fluid flows are simulated on the basis of the Navier-Stokes equations (1) – (2) (where S = 0 and p is the pressure) with one dimensionless parameter Re.
The spherical coordinate system R, θ, φ (x = R sinθ cosφ, y = R sinθ sinφ, z = R cosθ,
v = (vR, vθ, vφ)) and O-type grid are used. On the sphere surface the following boundary conditions have been used:
vR = vθ = vφ = 0, 0 2 1 2 / 2 / = ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ ∂ ∂ − ∂ ∂ = ∂ ∂ = =d R d R R x C R S R ρ
On the external boundary of the O-type grid the following boundary conditions have been used: 1) for z < 0: vR = cosθ, vθ = -sinθ, S = 0, vφ = 0; 2) for z ≥ 0: vR = cosθ, vθ = -sinθ, S = 0, ∂vφ/∂R = 0.
For solving of the equations (1) – (3) the Splitting on physical factors Method for Incompressible Fluid flows (SMIF-MERANGE) with hybrid explicit finite difference scheme (second-order accuracy in space, minimum scheme viscosity and dispersion, monotonous, capable for work in wide range of Reynolds and Froude numbers) based on Modified Central Difference Scheme (MCDS) and Modified Upwind Difference Scheme (MUDS) with special switch condition depending on velocity sign and sign of first and second differences of transferred functions has been used 12. The Poisson equation for the pressure has been solved by the Preconditioned Conjugate Gradients Method.
The parallelization of the algorithms has been made on the massive parallel computer with a distributed memory MBC 1000 (based on Intel Xeon processors (1.7 GHz)).
3 VISUALIZATION OF THE VORTICES
consider a local stream lines pattern around any point in a flow (where Im(σ1,2) > 0) in a reference frame x moving with the velocity of this point. It’s easy to demonstrate that the local stream lines pattern (v = dx/dt ≈ Vx) in the considered reference frame x is closed or spiral, and Im(σ1,2) is the angular velocity of this spiral motion.
4 HOMOGENEOUS FLUID FLOWS AROUND A SPHERE
Initially the steady axisymmetrical flows have been calculated here at any Re because of the axisymmetrical boundary conditions. Then some short-term perturbations have been introduced in the 3D axisymmetrical flows. At Re ≤ 200 these 3D axisymmetrical flows are stable with respect to flow perturbations. At Re > 200 these flow perturbations produce non-axisymmetrical flow regimes. At 200 < Re ≤ 270 the initially non-axisymmetrical vortex ring (in the recirculation zone) is deformed and after some time a steady double-thread wake is observed 1, 6-10. At Re > 270 the flow becomes unsteady and periodical (Figs. 1-7). The flow field can be divided into four parts: 1) the recirculation zone (Fig. 2 (right panel)), 2) the
vortex envelope (surrounding the recirculation zone) (Fig. 2-3), 3) the outer flow (surrounding
the vortex envelope and the vortex structures of the wake), 4) the vortex structures of the
wake (outgoing from the recirculation zone) (Fig. 3). As is well known from the experiments
at 280 < Re < 420 a mirror symmetry of the wake (Fig. 1) and periodical detachment of the top edge of the shear layer (surrounding the recirculation zone) have been observed 3, i.e. at 280 < Re < 420 one flow regime is realized. In the course of present investigations for the first time it was found that the detailed FMV for 270 < Re ≤ 290, 290 < Re ≤ 320 and 320 < Re ≤ 400 are different, i.e. at 270 < Re ≤ 400 three flow regimes have been observed. The main differences are observed in the recirculation zone (Figs. 2-5, 7).
a)
b)
c)
d)
Figure 3: Sphere near wake, Re = 280, isosurfaces Im(σ1,2) = 0.005 at y < 0 during the period: a) t = 1496.4,
a)
b)
c)
d)
Figure 4: Sphere near wake, Re = 280, instantaneous stream lines (left panel) and isolines Im(σ1,2) > 0 with step
At 270 < Re ≤ 290 a new vortex ring R1 is generated periodically in the recirculation zone (near the primary separation line, see Figs. 2, 3-4a) due to the Kelvin-Helmholtz instability. Then the top part of R1 is disappeared and generated again (Figs. 3-4b). Then the top part of R1 is connected with the vortex envelope (Figs. 3-4c) and replaces the disappearing main vortex ring of the recirculation zone (Figs. 3-4d). At 290 < Re ≤ 320 except R1 (Fig. 5a) an additional new vortex ring R2 is generated during the period (Fig. 5b). Then R2 induces the new vortex semi-ring R3 and is disappeared (Fig. 5c-d). Then R3 is connected with the vortex envelope (Fig. 5d) and replaces the disappearing main vortex ring of the recirculation zone. At 320 < Re ≤ 400 R1, R2 and R3 are generated and disappeared (Fig. 7a). Then the main vortex ring R0 is shifted closer to the sphere (Fig. 7a), connected with the vortex envelope (Fig. 7b) and shifted downstream (replaces itself).
a) c)
b) d)
Figure 5: Sphere near wake, Re = 320, isolines Im(σ1,2) > 0 with step 0.04 in the wake symmetry plane during
the period: a) t = 453, b) t = 455, c) t = 456.5, d) t = 460.
It was found that at the different parts of the flow the following six basic FMV can be selected in the detailed FMV during the period:
1k) the generation of a new vortex ring (or semi-ring) in the recirculation zone near the
sphere surface due to the Kelvin-Helmholtz instability (R1 in Figs. 2, 3-4a, 5a; R2 in Fig. 5b; R3 in Fig. 5c);
1f) the formation of the two vortex filaments (or threads, i.e. the legs of the nascent vortex
loop of the wake) connected with a new vortex ring in the recirculation zone (Fig. 3c-d);
1d) the displacement of the main vortex ring R0 (in the recirculation zone) closer to the
2s) the stretching of the side parts of the vortex envelope downstream (Fig. 3b-d);
2t/b) the top or bottom edge of the vortex envelope is rolling up cylindrically and detached
(Figs. 3-4 (c-d));
3t/b) the generation of the head of the vortex loop (facing upwards (t) or downwards (b))
in the outer flow (Fig. 3c-d and Fig. 3b-c correspondingly).
The number in the name of basic FMV means the part of the flow where this FMV works. The basic FMV 2s, t/b and 3t/b are caused by two vortex threads (the legs of the first vortex loop of the wake (outgoing from the recirculation zone)).
Thus the detailed FMV for different flow regimes are follows:
1) 270 < Re ≤ 290: {1k-2s-1k-3b}-{ 1f-3t-2t}; 2) 290 < Re ≤ 320: {1k-2s-1k-1k}-{2s- 1f-3t-2t}; 3) 320 < Re ≤ 400: {1k-2s-1k-1k}-{2s-1d-1f-3t-2t};
4) Re > 700: {2s-1d-1k-1f-3b-2b}-{2s-1d-1k-1f-3t-2t} (the braces select a half of the
period).
a)
b)
Figure 6: A procession of the vortex loops in the sphere wake, Re = 350, isosurfaces Im(σ1,2) = 0.05 during the
period: a) t = 882, b) t = 889.5.
a) b)
Figure 7: Sphere near wake, Re = 350, isolines Im(σ1,2) > 0 with step 0.2 in the wake symmetry plane:
At 270 < Re ≤ 360 a mirror symmetry of the wake is observed and the time-averaged lift/side and torque moment coefficients are not equal to zero. At 360 < Re ≤ 400 the slow regular rotation of the vortex envelope is observed (Strot = 0.0054, 0.0088, 0.0085, 0.0041 for
Re = 375, 380, 390, 400 correspondingly). At Re > 400 the vortex envelope is irregularly
rotated. At 270 < Re ≤ 400 the one (top) edge of the vortex envelope is rolling up and detached during the period. At Re > 700 the opposite edges of the vortex envelope are detached alternatively during the period. At 400 < Re < 700 the detailed FMV (3) and FMV (4) are realized disorderly one by one.
Re St <CD> <CL> Grid 280 0.133 0.683 0.079 1 290 0.140 0.675 0.082 1 300 0.145 0.669 0.084 1 320 0.148 0.654 0.086 1 350 0.133 0.627 0.068 2 360 0.150 0.630 0.084 1 375 0.183 0.622 0.082 1 380 0.183 0.619 0.083 1 390 0.144 0.615 0.081 1 400 0.133 0.603 0.059 3 500 0.08, 0.16 0.570 0.055 2 700 0.150 0.516 0.035 3
Table 1: Strouhal number St = fd/U (where f is the frequency of shedding of the vortices), the time-averaged drag <CD> and total lateral <CL> coefficients versus Re for grids: 1) 120 x 60 x 120, 2) 180 x 90 x 180,
3) 180 x 90 x 120.
At 5·104 ≤ Re ≤ 5·106 the monotonous reduction of the time-averaged total drag coefficient has been observed (from value 0.455 to 0.165) due to the laminar-turbulent transition in the boundary layer. It was shown that this drag crisis is appeared because of the formation of the separated bubbles within the boundary layer.
At Re ≤ 360 the obtained results (Table 1) are in a very good agreement with the papers1, 3, 7 and other experimental and numerical results. At 360 < Re < 3.7·105 an agreement is not so good.
5 STRATIFIED FLUID FLOWS AROUND A SPHERE
Owing to DNS of the flow induced by the diffusion around a resting sphere for the first
time it was shown that the interruption of the molecular flow (by the resting sphere) not only
generates the flow on the sphere surface but also creates the short unsteady internal waves 14. At first a number of these waves is equal to a number of buoyancy periods Tb (past from the beginning of the interruption of the molecular flow by the resting sphere (d = 2 cm,
by the radius of the sphere, is located near the sphere surface. At time more than 37·Tb the sizes and arrangement of cells are stabilized, and only the base cell and two thin adjacent cells with a thickness 2.2 mm are observed both in the salinity perturbation field S and in stream lines pattern (Fig. 8b). In other words the high gradient sheets of density near the poles of the resting sphere are observed in the left panel of Fig. 8b (the darker isolines correspond to the negative values of S). The free singular point of the base cell in the right panel of Fig. 8b is located at R = 1.11 cm and θ ≈ 37°. The thicknesses of the velocity and density boundary layers are equal to 0.029 cm. The maximum scalar of the velocity (5·10-4 cm/s) has been observed at the periphery of the boundary layer. Therefore for the diffusion-induced flow around the resting sphere (d = 2 cm, Tb = 6.34 s) it is possible to say about Fr = 2.5·10-4 and
Re = 0.1 (C = 500, ν = 0.01 cm2/s).
a)
b)
a) b)
c) d)
e) f) g)
Figure 9: Sphere wake, Fr = 0.02, Re = 50, C = 200: a-b) isosurfaces Im(σ1,2) = 0.005; c-d) stream lines in
vertical (x-z) (c) and horizontal (y-z) (d) planes; e) isolines of the salinity perturbation S with step 2·10-8 in
vertical (x-z) plane; f-g) skin friction patterns on the sphere top (f) and lee (g) sides.
The similar high gradient sheets of density have been observed before a moving sphere (near the poles) at Fr ≤ 0.02 (Figs. 9a-b, 9e). In Fig. 9 the flow regime “Two steady
two-dimensional attached vortices with vertical rotation axes in the recirculation zone” (Fr < 0.2, Re < 120) is demonstrated. The flow at Fig. 9a-b can be divided into eight parts: 1) the
observed (Figs. 9f-g). At 0.05 ≤ Fr ≤ 0.2 the ring-like primary separation line and lee waves with wavelength λ/d ≈ 2π·Fr have been observed.
The flow regime “Unsteady two-dimensional vortices with vertical rotation axes” (Fr < 0.15, Re > 120) is demonstrated in Fig. 10 (Fr = 0.05, Re = 500, St1 = 0.24, St2 = 0.11).
a) b) c)
d) e)
Figure 10: Sphere wake, Fr = 0.05, Re = 500, C = 100: a-b) skin friction patterns on sphere top (a) and lee (b) sides; c) isosurface Im(σ1,2) = 0.2; d-e) instantaneous stream lines in vertical (d) and horizontal (e) planes.
Fr Re ∆CD θH θV L/d 0.004 10 1.685 0.3° 84.0° 0.03 0.005 100 0.582 63.3° 80.2° 2.22 0.02 50 0.720 53.0° 72.5° 1.19 0.05 500 1.800 --- 63.2° --- 0.08 100 0.655 67.9° 55.9° 2.36 0.3 750 0.600 39.2° 15.0° --- 0.4 1000 0.600 70.0° 10.0° --- 0.5 100 0.588 0.0° 0.0° 0.0 0.6 100 0.301 0.0° 0.0° 0.0 0.7 100 0.091 59.1° 20.6° --- 1.0 200 0.015 74.2° 28.0° 0.52
Table 2: Modified drag coefficient ∆CD(Fr, Re) = CD(Fr, Re) - CD(∞, Re); horizontal θH and vertical θV
separation angles and recirculation zone length L/d (measured from the rear stagnation point) versus (Fr, Re) (grid: 120 x 60 x 120).
with vertical rotation axes near y-z plane are also presented here (Fig. 11e, St = 0.2). The obtained Strouhal numbers are in a good agreement with 11 where for 0.03 < Fr < 0.28 and 120 < Re < 500 St is equal to 0.2.
Thus at Fr ≤ 0.4 the wake has a quasi-two-dimensional structure near horizontal plane y-z. While at Fr = 1 the flow is fully 3D (Figure 12, Fr = 1, Re = 200, the flow regime
“Non-axisymmetric attached vortex in the recirculation zone” (Fr > 0.4, Re < 500)). The oval
primary separation line with eight singular points (four saddles (in x-z and y-z planes) and four foci (near y-z plane)) has been observed both in Fig. 11c (Fr = 0.3) and in Fig. 12c (Fr = 1). At 0.4 ≤ Fr ≤ 1 the form of the primary separation line is strongly changed. At 0.5 ≤ Fr ≤ 0.6 the primary separation line is disappeared (transformed into the rear stagnation point). At
Fr = 0.7 the oval primary separation line with eight singular points have been observed but
instead of the node in the rear stagnation point the saddle (in the rear stagnation point) and two nodes (in y-z plane not far from saddle points of the primary separation line) have been found. The forward vortex tubes in Figs. 11a and 12a (the part of the wave pattern) form a first vortex row. The angle 2·ψ between the left (y > 0) and right (y < 0) branches of the forward vortex tubes (Figs. 12a) is reduced with Froude number increasing (ψ = 58°, ψ = 55°, ψ = 33°, ψ = 23° for Fr = 0.3, 0.5, 0.7, 1.0 correspondingly).
The results obtained in this work for stratified fluid (Table 2) are in a good agreement with11 and other experimental and numerical results.
a) b) c)
d) e)
Figure 11: Sphere wake, Fr = 0.3, Re = 750, C = 200: a) isosurface Im(σ1,2) = 0.02; b-c) skin friction patterns on
6 CONCLUSIONS
- The homogeneous (1 ≤ Re ≤ 5·106) and stratified (0.004 ≤ Fr ≤ 1, 10 ≤ Re ≤ 1000) viscous incompressible fluid flows around a sphere have been investigated by means of DNS and the visualization of the vortex structures in the wake.
- In case of the homogeneous fluid at 200 < Re ≤ 1000 the detailed formation mechanisms of vortices (FMV) in the sphere wake have been described for the different unsteady periodical flow regimes (270 < Re ≤ 290, 290 < Re ≤ 320, 320 < Re ≤ 400, 400 < Re < 700 and Re > 700) and the six basic FMV have been selected; at 290 < Re ≤ 320 a new flow regime has been discovered; at 5·104 ≤ Re ≤ 5·106 the monotonous reduction of the time-averaged total drag coefficient has been observed (from value 0.455 to 0.165).
- In case of the stratified fluid the four different flow regimes have been simulated; the high gradient sheets of density have been observed near the poles of the resting sphere and the moving sphere (Fr ≤ 0.02). For the first time the complex 3D vortex structures of these flows have been observed.
a) b) c)
d) e)
Figure 12: Sphere wake, Fr = 1, Re = 200, C = 9216.6: a) isosurface Im(σ1,2) = 0.02; b-c) skin friction patterns
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