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A computational study of coherent wake structures behind 2-D bluff bodies
Journal c!f WbM Engineering and Industrial Aerodynamics, 41-44 (1992) 2853-2861
2853
Elsevier
A C o m p u t a t i o n a l S t u d y of C o h e r e n t Wake S t r u c t u r e s B e h i n d 2 - D Bluff B o d i e s J. A. P e a r c e " , All Q a s i m u, T. T. M a x w e l l c, a n d S. P a r a m e s a w a r a n d " Captain, United States Air Force, Wright Patterson AFB, Ohio b Graduate Research Assistant, Mechanical Engineering Department, Texas Tech University, Lubbock, Texas c Associate Professor, Mechanical Engineering Department, Texas Tech University, Lubbock, Texas d Assistant Professor, Mechanical Engineering Department, Texas Tech University, Lubbock, Texas ABSTRACT
The periodic shedding of vortices from bluffbody geometries was predicted with a 2-D finite--difference code. Two geometrical configurations were considered: a flat plate in uniform flow and a fiat plate near a ground plane. Strouhal numbers and aerodynamic force coefficients were computed for all geometries considered and these results compare well with experimental results from the literature. The results indicate that the code is capable of predicting periodic vortex shedding for the geometries tested. 1.
INTRODUCTION
Vortex shedding is important in the design of structures that will be subjected to wind or water flow loadings. Of more recent interest is the effect of vortex shedding on the stability of motor vehicles. In general, any bluffbody subjected to fluid flow can potentially shed vortices; knowledge of the frequency and amplitude of the associated loading can be vital to stability and structural integrity of the body. The primary objective of this paper is to demonstrate the ability of a finite-difference code to predict periodic vortex shedding. Two basic flow geometries are considered: a fiat plate in a uniform flow and a flat plate near a ground plane in a uniform flow as shown in Figure 1. 2.
BACKGROUND
At low Reynolds numbers (Re < 60) a stationary, symmetric vortical pattern is formed behind a bluff body. At higher Reynolds numbers a periodic wake is formed. 0167-6105/92/S05.(X)© 1992ElsevierSciencePublishersB.V. All rights reserved.
2854 Wake formation is described by Younis [1] and Roshko [2, 3, and 4]. The flow separates from the body's leading edges creating free shear layers that continue downstream before rolling into vortices. This flow pattern is inherently unstable with vortices being alternately shed from the top and bottom surfaces of the body to form the familiar Karman vortex street. Roshko noted that the bluffness of a body is related to the ratio of wake width to body size. A bluffer body diverges the flow more extensively producing a wider wake and a lower shedding frequency. Roshko determined that the Strouhal number for flow over a flat plate remained nearly constant (0.130 < S t < 0.140) over a wide range of Reynolds number and he devised a method to determine the drag on a flat plate from the measured base pressure coefficient.
Flate Plate in a Uniform Flow
;~= l/h
h
Flate Plate Near a Ground Plane in a Uniform Flow
Figure 1 F l o w Geometries
Matty [5] and Strickland et al. [6] studied the effects of vortex shedding from heliostats. Though the actual geometry was not a flat plate, they performed wind tunnel tests on a flat plate near a ground plane. Shedding frequencies were determined for the 3-D flat plate. From these data, an equation was derived t h a t approximates the Strouhal number in terms of R e and ~ where ~ is the ratio of distance from the wall to plate height.
3.
COMPUTATIONAL MODEL
The code used in this work is based on Spalding and Patankar's Semi-Implicit Method for Pressure Linked Equations or SIMPLE [7]. The code will handle 2-D, incompressible, unsteady, elliptic flows. For most of the computations presented the flow was assumed to be laminar; however, for comparison the k - e turbulence model embodied in the code was used in some of the predictions. Both uniformly spaced and nonuniformly spaced Cartesian grids were used in the computations. The shedding frequency is computed from the numerical results by examining the periodic fluctuations in three fundamental quantities as did Younis. The frequency n is determined by examining the axial velocity at two locations behind the body, the drag coefficient C D, and ~1 the lift coefficient C L. The flow field must be allowed to reach a cyclic state before the frequency can be determined. Younis showed that the plate's drag coefficient oscillates at twice the shedding frequency while the lift coefficient oscillates at the shedding frequency.
2855 4.
RESULTS
4.1 V o r t e x S h e d d i n g F r o m a Flat P l a t e Vortex shedding from a fiat plate was predicted using both a 60 by 50 uniformly spaced grid and a 150 by 80 nonuniform grid. On the uniform grid the fiat plate was 0.006 m or 10 cells long and was centered vertically within the 0.06 m long by 0.03 m high calculation domain. The plate was located approximately two plate lengths from the top and bottom of the grid, 2.5 plate lengths from the inlet boundary, and 7.5 plate lengths from the outlet of the domain. For the nonuniform grid the 0.0102 m long plate was composed of 16 equally spaced cells and was centered vertically within the 0.175 m long by 0.140 m high calculation domain. There were approximately 6.4 plate lengths between the plate and the top and bottom of the grid. The inlet boundary was two plate lengths upstream of the plate and the outlet was approximately 15 plate lengths downstream of the plate. For both grids the upper and lower boundaries were slip walls or symmetry boundaries. A uniform inlet flow was directed normal to the plate. Inlet velocities from 0.3 m/s to 3.0 m/s were used to achieve a range of Reynolds numbers of approximately 2,000 < R e > 30,000. Geometrically, the flow is symmetric about the horizontal line which bisects the flat plate. To allow the periodic vortex shedding to develop it was necessary to add an initial perturbation to the flow field. A momentum source was included for the first three time steps and was located at a control volume directly behind and at the upper limit of the plate. Predictions of the Karman vortex street are clearly demonstrated by Figures 2---6. These figures show the absolute velocities for equally separated increments in time. The five plots progress by steps of a quarter of the vortex shedding period, P, to show the progression of the vector field with time. Absolute velocities are displayed because the vortex shedding pattern is more apparent when shown in this manner. The absolute velocity is computed from the relation V~b~= Vm~ - U and it represent the movement of the plate through the fluid rather than the movement of the fluid past the plate. At t = 0, the familiar alternating vortex pattern of the vortex street is clearly visible. There is a pronounced spreading of the vortices in the vertical direction and growth of the vortices as they move downstream. These characteristics indicate that the finite-difference code accurately predicts the vortex shedding pattern. Progressing to t = 1/4 P, notice that the entire flow pattern from t = 0 has moved downstream. There is a disturbance at the top of the plate which is the beginning of the formation of the next vortex. At t = 1/2 P, the vortex on the top of the plate has formed and is clearly recognizable. As expected, at t = 1/2 P the velocity field appears to be a mirror image about the horizontal bisector of the field at t = 0. The velocity field at t= 3/4 P indicates the beginning of the formation of a new vortex at the bottom of the plate. Finally, at t = P the velocity field appears to be the same as calculated for t = 0. The velocity field has cyclically progressed back to its initial
2856
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....
, ,,,
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..
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' •
:.:~ .............
~
,
,
,
~
-
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,..
.,
F i g u r e 5 V e l o c i t y F i e l d at t ~ = 314 P
-..- ~.
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F i g u r e 3 V e l o c i t y F i e l d at Iref = 114 P , ~. . . . . ,~l//.~,.\\\.~)tt~t0,,-, ~ ~ I t ! z~-~ • ,, z--..~ ;
, , , . . . ~ , . . . . , , , ~
*~
;
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.
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!
Ill F i g u r e 6 V e l o c i t y F i e l d at Iref = P
state, and the predicted vortex shedding is indeed periodic. Among the most critical parameters associated with periodic vortex shedding is the Strouhal number. Roshko provided extensive data on the Strouhal number for various geometries, including the flat plate. Figure 7 shows a quantitative comparison of the experimental results of Roshko with current predictions.
Examination of Figure 7 indicates that the predicted Strouhal numbers are comparable to those measured by Roshko. For the 150 by 80 grid, prediction of the Strouhal number was accurate up to a Reynolds number of 10t For Reynolds numbers greater t han 10 ~, the predicted Strouhal numbers dropped below Roshko's measurements. The error between the predicted and experimental results did not exceed 15%. The coarser grid predicted Strouhal numbers more consistent with Roshko's results over the entire Reynolds number range. Strouhal numbers predicted with the coarse grid were generally greater than those predicted with the fine grid. Addition of the k-e turbulence model caused a slight increase in predicted
2857 Strouhal number. The results were favorable w hen c o m p a r e d with Younis' work. Younis predicted Strouhal numbers for a flat plate in a range of 0.165--0.185 which appear to be an o v e r e s t i m a t e of t h e Strouhal number when compared with the exp e r i m e n t a l d a t a provided by Roshko.
0.20
0.15 OI
o
0 0.10 El RoBhko (1954) 0 150 x 80 grid •
0.05
60 x 80 grid
• 60 x 80 grid with k--emodel
Roshko provided experimental results for the Reynolds Number time averaged pressures along the center line of Figure 7 Strouhal Number versus Reynolds the wake. This data, for Number for a Flat Plate Re = 14,500, was given in terms of Cp and x / l . The quant i t yx is measured from the plate with positive x extending downstream. A comparison with Roshko's data was made for both grid sizes and for the case where turbulence was included. Figures 8 and 9 present the variation in C with x / l for the fine and course grids. The most obvious difference between the computations and experimental results is the magnitude of the base pressure or suction just downstream of the plate. The predictions are nearly twice as large as Roshko's measurements. The predicted location and magnitude of the minimum pressure were also quite different from the experimental results. However, this difference is not necessarily the fault of the computational model, because Roshko cited difficulties 0.0 in obtaining the wake pressure measurements. --0.5 ~ The anomalies cited for - 1.o ./"~" the fine grid solution were also a p p a r e n t in t h e / coarse grid solution. The ® - 1.5 / p r i m a r y difference be- 2.0 ] tween the solutions for Roshko (1954) / ..... 150 x 80 grid the two grid lies in the k\ . / / accuracy of the solution - 3.0 -for values of x / l > 2.0 where the fine grid soluI I I I 0.0 2.0 4.o 6.o 8.o lO.O tion was much closer to x/l the experimental results than was the coarse grid F i g u r e 8 W a k e C e n t e r L i n e P r e s s u r e D i s t r i b u t i o n for 150 x 80 grid a t R e = 14,500 solution. 0,0~
i
103
i
i
i
i
I
I
i
i
104
i
i
i
2858 Calculations of the time averaged lift and drag coefficients were carried out for each case run. As expected, the average lift coefficient CL~_~was zero for all runs though instantaneous values of lift were nonzero. The zero value of lift was expected since this case is for a flat plate at zero angle of attack.
0.0
-
0.5
-
1.0
-
1.5
/ - 2.0
~
/
/./" ~-~
Ro6hko (1964) 60 x 50 grid
- 2.5 -3.0
0.0
I
I
I
2.0
4.0
6.0
8.0
x/l The calculated average d r a g coefficients are F i g u r e 9 W a k e C e n t e r L i n e P r e s s u r e D i s t r i b u t i o n shown in Figure 10 with f o r 60 x 50 g r i d a t R e = 14,500 the drag coefficients derived from Roshko's data. The predicted values of C D exceed Roshko's values by more than 50%, which may well be tied to the much lower predicted base pressures. However, also note t h a t the predicted values agree favorably with the numerical results of Younis who found CD's in the range of 3,25--3.65.
There were significant differences in the predicted drag coefficients. The fine grid produced drag coefficients 10% greater than did the coarse grid. This fact may be related to the lower minimum wake pressures predicted with the fine grid. Including turbulence in the solution increased the predicted drag coefficient by about 1%. 4.2 V o r t e x S h e d d i n g f r o m a F l a t P l a t e N e a r a G r o u n d P l a n e
The numerical prediction of vortex shedding from a fiat plate near a ground plane was carried out on a uniform 80 by 50 grid with dimensions of 0.08 m by 0,03 m. The plate was composed of 12 vertical cells and was 0.0072 m high. The inlet boundary was located two plate heights upstream, and the exit was located nine plate heights downstream. The location of the bottom boundary relative to the plate was governed by ~ which was assigned the values of 0, 1/4, 1/2, and 1. The spacing between the plate and the upper boundary varied between 2 and 3 plate heightsfor different values of~. Boundary conditions were similar to those used for the flat plate predictions except the bottom boundary was made an impermeable wall. The inlet velocity was varied between 2.0 m/s and 20.0 m/s to give a range of Reynolds number of approximately 1,000 < R e > 15,000. This range of Reynolds number was chosen to match the range examined by Matty. Periodic flow was initiated in the same manner as described for the flat plate. Inducing vortex shedding in the wake of a flat plate near a ground plane was not
2859
3.5
0 3.0
0
•
0
i
•
o •
O0 W
l
o
e|
2.5 2.0
m
1.5
m
1.0
m
0.5
i
[] ~o [] @ ~ o o o o B o o
oo
[] Roshko (1954) O 150 x 80 grid •
60 x 50 grid
•
60 x 50 grid with k-e model
I
I
I
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I
IIII
successful for all cases. To achieve vortex shedding, the plate was moved one plate length from the wall where shedding occurred readily. To achieve shedding for other values of ~., the velocity field of the £ = 1 case was imposed as an initial condition.
I
Predicted Strouhal numberversus Reynolds numReynolds Number ber for various values of/~ F i g u r e 10 C . v e r s u s R e y n o l d s N u m b e r is shown in Figure 11. For for a Flat Plate ~= 1, the greatest separation between the plate and the ground plane, the prediction of Strouhal number for the entire range of Reynolds number was accurate. As the Reynolds number was increased, the numerical solution tended to give a slightly higher value of the Strouhal number than did Matty's function. However, the Strouhal number generally decreased for increasing Reynolds number as indicated by Matty's study. 0.0
lO 3
lo 4
For ~ = 1/2, predicted values of the Strouhal number were lower than those given by Matty's function for low values of Reynolds number. However, as the Reynolds number increased the predic0.20 tions became more accurate. T he t r e n d of d e c r e a s i n g Strouhal number with increasing Reynolds number was also 0.15 supported by the prediction. The results were also similar for ;t = 1/4, though predicted 0.10 ~" Strouhal numbers slightly unm derestimated the empirical re• ,~=1 lation. Nonetheless, the trend ~ = 1/2 of decreasing Strouhal num0.05 • ;t = 1 / 4 ber with increasing Reynolds m number was again supported. I i I ]iilil f The variation in the Strouhal 0.00 104 105 number with)t for a fixed value of the Reynolds number was Reynolds Number also examined. Strouhal numF i g u r e 11 S t r o u h a l N u m b e r v e r s u s bers for 2 = 1/4 were generally Reynolds Number for a Flat greater than those for ;t = 1/2. P l a t e N e a r a Wall Matty's data indicate a simi-
2860 lar behavior between these values of A. However, the predicted Strouhal numbers at A = 1 were significantly higher than those at A = 1/2, though Matty's study indicates that these values should be nearly equal. The difference here m a y well liein the fact that Matty's study was for a 3D plate, while the predictions were for a 2-D plate.
5.0
4.5
4.0
3.5-
3.0 0.00
I
I
I
0.25
0.50
0,75
1.00
A
The predicted aerodynamic F i g m ~ 12 D ~ Coefficient versus forces changed with A, but Ground Clearance, A as expected for a bluffbody, there was little dependence of the drag on Reynolds number. Figure 12 presents the variation in drag coefficient with ;t. The trend for the drag to increase as ;t decreases is dearly apparent. At ;t = 0, the drag attained a value about 50% greater than t h a t obtained for a flat plate without a ground plane. This behavior was expected, for the increase in the drag on a body as it approaches a ground plane is well documented. The lift also increased as the ground clearance decreased, though the trend was not as prominent as for the drag. The predicted lift coefficient tended to decrease as Reynolds number increased. However, it must be noted that the magnitude o f the lift coefficient was quite small, and the lift may be considered to be i n s i ~ f i c a n t . 5.
CONCLUSIONS
This study proposed to demonstrate the ability to predict the phenomena of vortex shedding from 2-D bluffbodies. For every case tested with the exception of a fiat plate in contact with a ground plane, periodic vortex shedding was achieved. For the initial case involving a fiat plate in a uniform flow the results were favorable. Predicted Strouhal numbers agreed well with experimental data. Predicted pressures in the wake of the plate were much lower than the experimental data; however, the difference may be due to difficulties in experimentally measuring the pressures. Predictions of the drag also indicated deviations from the drag function provided by Roshko, but the accuracy of Roshko's function was unknown. With respect to parameters affecting the solution of the finite-difference equations, some significant results were discovered. Refinement of the finite-difference grid yielded only a slight improvement in the prediction of pressure distributions, and actually hampered the prediction of Strouhal numbers. The inclusion ofturbulence was found to have very little effect.
2861 The prediction of vortex shedding from a flat plate near a ground plane also yielded favorable results. The comparison of predicted Strouhal numbers with an empirically derived function demonstrated the ability to predict general trends in the shedding frequency. Quantitatively, there were significant differences in the prediction and the empirical function; however, this may be attributable to the fact that the prediction was 2-D while the function was derived for a 3-D case. Also, the expected increase in the aerodynamic force coefficients as the body approached the ground plane were accurately predicted. 6.
ACKNOWLEDGMENTS
The authors would like to thank Mr. Larry Socha and Mr. Jack Williams of Aerodynamics Department at Ford Motor Company for their help and support during the study. 7.
REFERENCES
1. Younis, B. Ain Lami (1988). On Modeling the Vortex Shedding From Bluff Bodies in Laminar and Turbulent Streams, Presented at the Seventh International Conference on Offshore Mechanics and Arctic Engineering, Houston, TX, February 7-i2, 1988. 2. Roshko, Anatol (1953). On the Development of Turbulent Wakes From Vortex Streets, NACA T e c h n i c a l N o t e 2913. 3. Roshko, Anatol (1954). A New Hodograph for Free-Streamline Theory, NACA T e c h n i c a l N o t e 316K 4. Roshko, Anatol (1954). On the Drag and Shedding Frequency of Two-Dimensional Bluff Bodies, NACA T e c h n i c a l Note 3169. 5. Matty, Rosemary R. (1979). V o r t e x S h e d d i n g F r o m S q u a r e P l a t e s N e a r a G r o u n d P l a n e : An E x p e r i m e n t a l S t u d y , M. S. Thesis, Texas Tech University. 6. Strickland, James H., Matty, Rosemary R., and Barton, Gregory H (1980); Vortex Shedding From Square Plates Perpendicular to a Ground Plane, AIAA J o u r n a l 18, pp. 715-716. 7. Patankar, Suhas V. (1980). N u m e r i c a l H e a t T r a n s f e r a n d F l u i d Flow. Hemisphere Publishing Corporation, New York, NY.
2853
Elsevier
A C o m p u t a t i o n a l S t u d y of C o h e r e n t Wake S t r u c t u r e s B e h i n d 2 - D Bluff B o d i e s J. A. P e a r c e " , All Q a s i m u, T. T. M a x w e l l c, a n d S. P a r a m e s a w a r a n d " Captain, United States Air Force, Wright Patterson AFB, Ohio b Graduate Research Assistant, Mechanical Engineering Department, Texas Tech University, Lubbock, Texas c Associate Professor, Mechanical Engineering Department, Texas Tech University, Lubbock, Texas d Assistant Professor, Mechanical Engineering Department, Texas Tech University, Lubbock, Texas ABSTRACT
The periodic shedding of vortices from bluffbody geometries was predicted with a 2-D finite--difference code. Two geometrical configurations were considered: a flat plate in uniform flow and a fiat plate near a ground plane. Strouhal numbers and aerodynamic force coefficients were computed for all geometries considered and these results compare well with experimental results from the literature. The results indicate that the code is capable of predicting periodic vortex shedding for the geometries tested. 1.
INTRODUCTION
Vortex shedding is important in the design of structures that will be subjected to wind or water flow loadings. Of more recent interest is the effect of vortex shedding on the stability of motor vehicles. In general, any bluffbody subjected to fluid flow can potentially shed vortices; knowledge of the frequency and amplitude of the associated loading can be vital to stability and structural integrity of the body. The primary objective of this paper is to demonstrate the ability of a finite-difference code to predict periodic vortex shedding. Two basic flow geometries are considered: a fiat plate in a uniform flow and a flat plate near a ground plane in a uniform flow as shown in Figure 1. 2.
BACKGROUND
At low Reynolds numbers (Re < 60) a stationary, symmetric vortical pattern is formed behind a bluff body. At higher Reynolds numbers a periodic wake is formed. 0167-6105/92/S05.(X)© 1992ElsevierSciencePublishersB.V. All rights reserved.
2854 Wake formation is described by Younis [1] and Roshko [2, 3, and 4]. The flow separates from the body's leading edges creating free shear layers that continue downstream before rolling into vortices. This flow pattern is inherently unstable with vortices being alternately shed from the top and bottom surfaces of the body to form the familiar Karman vortex street. Roshko noted that the bluffness of a body is related to the ratio of wake width to body size. A bluffer body diverges the flow more extensively producing a wider wake and a lower shedding frequency. Roshko determined that the Strouhal number for flow over a flat plate remained nearly constant (0.130 < S t < 0.140) over a wide range of Reynolds number and he devised a method to determine the drag on a flat plate from the measured base pressure coefficient.
Flate Plate in a Uniform Flow
;~= l/h
h
Flate Plate Near a Ground Plane in a Uniform Flow
Figure 1 F l o w Geometries
Matty [5] and Strickland et al. [6] studied the effects of vortex shedding from heliostats. Though the actual geometry was not a flat plate, they performed wind tunnel tests on a flat plate near a ground plane. Shedding frequencies were determined for the 3-D flat plate. From these data, an equation was derived t h a t approximates the Strouhal number in terms of R e and ~ where ~ is the ratio of distance from the wall to plate height.
3.
COMPUTATIONAL MODEL
The code used in this work is based on Spalding and Patankar's Semi-Implicit Method for Pressure Linked Equations or SIMPLE [7]. The code will handle 2-D, incompressible, unsteady, elliptic flows. For most of the computations presented the flow was assumed to be laminar; however, for comparison the k - e turbulence model embodied in the code was used in some of the predictions. Both uniformly spaced and nonuniformly spaced Cartesian grids were used in the computations. The shedding frequency is computed from the numerical results by examining the periodic fluctuations in three fundamental quantities as did Younis. The frequency n is determined by examining the axial velocity at two locations behind the body, the drag coefficient C D, and ~1 the lift coefficient C L. The flow field must be allowed to reach a cyclic state before the frequency can be determined. Younis showed that the plate's drag coefficient oscillates at twice the shedding frequency while the lift coefficient oscillates at the shedding frequency.
2855 4.
RESULTS
4.1 V o r t e x S h e d d i n g F r o m a Flat P l a t e Vortex shedding from a fiat plate was predicted using both a 60 by 50 uniformly spaced grid and a 150 by 80 nonuniform grid. On the uniform grid the fiat plate was 0.006 m or 10 cells long and was centered vertically within the 0.06 m long by 0.03 m high calculation domain. The plate was located approximately two plate lengths from the top and bottom of the grid, 2.5 plate lengths from the inlet boundary, and 7.5 plate lengths from the outlet of the domain. For the nonuniform grid the 0.0102 m long plate was composed of 16 equally spaced cells and was centered vertically within the 0.175 m long by 0.140 m high calculation domain. There were approximately 6.4 plate lengths between the plate and the top and bottom of the grid. The inlet boundary was two plate lengths upstream of the plate and the outlet was approximately 15 plate lengths downstream of the plate. For both grids the upper and lower boundaries were slip walls or symmetry boundaries. A uniform inlet flow was directed normal to the plate. Inlet velocities from 0.3 m/s to 3.0 m/s were used to achieve a range of Reynolds numbers of approximately 2,000 < R e > 30,000. Geometrically, the flow is symmetric about the horizontal line which bisects the flat plate. To allow the periodic vortex shedding to develop it was necessary to add an initial perturbation to the flow field. A momentum source was included for the first three time steps and was located at a control volume directly behind and at the upper limit of the plate. Predictions of the Karman vortex street are clearly demonstrated by Figures 2---6. These figures show the absolute velocities for equally separated increments in time. The five plots progress by steps of a quarter of the vortex shedding period, P, to show the progression of the vector field with time. Absolute velocities are displayed because the vortex shedding pattern is more apparent when shown in this manner. The absolute velocity is computed from the relation V~b~= Vm~ - U and it represent the movement of the plate through the fluid rather than the movement of the fluid past the plate. At t = 0, the familiar alternating vortex pattern of the vortex street is clearly visible. There is a pronounced spreading of the vortices in the vertical direction and growth of the vortices as they move downstream. These characteristics indicate that the finite-difference code accurately predicts the vortex shedding pattern. Progressing to t = 1/4 P, notice that the entire flow pattern from t = 0 has moved downstream. There is a disturbance at the top of the plate which is the beginning of the formation of the next vortex. At t = 1/2 P, the vortex on the top of the plate has formed and is clearly recognizable. As expected, at t = 1/2 P the velocity field appears to be a mirror image about the horizontal bisector of the field at t = 0. The velocity field at t= 3/4 P indicates the beginning of the formation of a new vortex at the bottom of the plate. Finally, at t = P the velocity field appears to be the same as calculated for t = 0. The velocity field has cyclically progressed back to its initial
2856
!! ,, .... ,,,,,.......,,,,,,,,,,
....
, ,,,
F i g u r e 2 V e l o c i t y F i e l d at t ~ = 0
..
~
,
. . . . . . . .
' •
:.:~ .............
~
,
,
,
~
-
~ _ _ -
,..
.,
F i g u r e 5 V e l o c i t y F i e l d at t ~ = 314 P
-..- ~.
l]i
- - - , - , - , , , , , ~
I1~I;!Iii1111
-" ; ~,
F i g u r e 3 V e l o c i t y F i e l d at Iref = 114 P , ~. . . . . ,~l//.~,.\\\.~)tt~t0,,-, ~ ~ I t ! z~-~ • ,, z--..~ ;
, , , . . . ~ , . . . . , , , ~
*~
;
"~ .~
~ ~,,-.---~,-,'~/t|}tr~--,,'
.
/ t ~ ~ ~
, ~--~-,, ..... ,~t~'..-.'; ~ ~ | ~ ~ F i g u r e 4 V e l o c i t y F i e l d at t f = 1/2 P
!
Ill F i g u r e 6 V e l o c i t y F i e l d at Iref = P
state, and the predicted vortex shedding is indeed periodic. Among the most critical parameters associated with periodic vortex shedding is the Strouhal number. Roshko provided extensive data on the Strouhal number for various geometries, including the flat plate. Figure 7 shows a quantitative comparison of the experimental results of Roshko with current predictions.
Examination of Figure 7 indicates that the predicted Strouhal numbers are comparable to those measured by Roshko. For the 150 by 80 grid, prediction of the Strouhal number was accurate up to a Reynolds number of 10t For Reynolds numbers greater t han 10 ~, the predicted Strouhal numbers dropped below Roshko's measurements. The error between the predicted and experimental results did not exceed 15%. The coarser grid predicted Strouhal numbers more consistent with Roshko's results over the entire Reynolds number range. Strouhal numbers predicted with the coarse grid were generally greater than those predicted with the fine grid. Addition of the k-e turbulence model caused a slight increase in predicted
2857 Strouhal number. The results were favorable w hen c o m p a r e d with Younis' work. Younis predicted Strouhal numbers for a flat plate in a range of 0.165--0.185 which appear to be an o v e r e s t i m a t e of t h e Strouhal number when compared with the exp e r i m e n t a l d a t a provided by Roshko.
0.20
0.15 OI
o
0 0.10 El RoBhko (1954) 0 150 x 80 grid •
0.05
60 x 80 grid
• 60 x 80 grid with k--emodel
Roshko provided experimental results for the Reynolds Number time averaged pressures along the center line of Figure 7 Strouhal Number versus Reynolds the wake. This data, for Number for a Flat Plate Re = 14,500, was given in terms of Cp and x / l . The quant i t yx is measured from the plate with positive x extending downstream. A comparison with Roshko's data was made for both grid sizes and for the case where turbulence was included. Figures 8 and 9 present the variation in C with x / l for the fine and course grids. The most obvious difference between the computations and experimental results is the magnitude of the base pressure or suction just downstream of the plate. The predictions are nearly twice as large as Roshko's measurements. The predicted location and magnitude of the minimum pressure were also quite different from the experimental results. However, this difference is not necessarily the fault of the computational model, because Roshko cited difficulties 0.0 in obtaining the wake pressure measurements. --0.5 ~ The anomalies cited for - 1.o ./"~" the fine grid solution were also a p p a r e n t in t h e / coarse grid solution. The ® - 1.5 / p r i m a r y difference be- 2.0 ] tween the solutions for Roshko (1954) / ..... 150 x 80 grid the two grid lies in the k\ . / / accuracy of the solution - 3.0 -for values of x / l > 2.0 where the fine grid soluI I I I 0.0 2.0 4.o 6.o 8.o lO.O tion was much closer to x/l the experimental results than was the coarse grid F i g u r e 8 W a k e C e n t e r L i n e P r e s s u r e D i s t r i b u t i o n for 150 x 80 grid a t R e = 14,500 solution. 0,0~
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103
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2858 Calculations of the time averaged lift and drag coefficients were carried out for each case run. As expected, the average lift coefficient CL~_~was zero for all runs though instantaneous values of lift were nonzero. The zero value of lift was expected since this case is for a flat plate at zero angle of attack.
0.0
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Ro6hko (1964) 60 x 50 grid
- 2.5 -3.0
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x/l The calculated average d r a g coefficients are F i g u r e 9 W a k e C e n t e r L i n e P r e s s u r e D i s t r i b u t i o n shown in Figure 10 with f o r 60 x 50 g r i d a t R e = 14,500 the drag coefficients derived from Roshko's data. The predicted values of C D exceed Roshko's values by more than 50%, which may well be tied to the much lower predicted base pressures. However, also note t h a t the predicted values agree favorably with the numerical results of Younis who found CD's in the range of 3,25--3.65.
There were significant differences in the predicted drag coefficients. The fine grid produced drag coefficients 10% greater than did the coarse grid. This fact may be related to the lower minimum wake pressures predicted with the fine grid. Including turbulence in the solution increased the predicted drag coefficient by about 1%. 4.2 V o r t e x S h e d d i n g f r o m a F l a t P l a t e N e a r a G r o u n d P l a n e
The numerical prediction of vortex shedding from a fiat plate near a ground plane was carried out on a uniform 80 by 50 grid with dimensions of 0.08 m by 0,03 m. The plate was composed of 12 vertical cells and was 0.0072 m high. The inlet boundary was located two plate heights upstream, and the exit was located nine plate heights downstream. The location of the bottom boundary relative to the plate was governed by ~ which was assigned the values of 0, 1/4, 1/2, and 1. The spacing between the plate and the upper boundary varied between 2 and 3 plate heightsfor different values of~. Boundary conditions were similar to those used for the flat plate predictions except the bottom boundary was made an impermeable wall. The inlet velocity was varied between 2.0 m/s and 20.0 m/s to give a range of Reynolds number of approximately 1,000 < R e > 15,000. This range of Reynolds number was chosen to match the range examined by Matty. Periodic flow was initiated in the same manner as described for the flat plate. Inducing vortex shedding in the wake of a flat plate near a ground plane was not
2859
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[] Roshko (1954) O 150 x 80 grid •
60 x 50 grid
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60 x 50 grid with k-e model
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successful for all cases. To achieve vortex shedding, the plate was moved one plate length from the wall where shedding occurred readily. To achieve shedding for other values of ~., the velocity field of the £ = 1 case was imposed as an initial condition.
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Predicted Strouhal numberversus Reynolds numReynolds Number ber for various values of/~ F i g u r e 10 C . v e r s u s R e y n o l d s N u m b e r is shown in Figure 11. For for a Flat Plate ~= 1, the greatest separation between the plate and the ground plane, the prediction of Strouhal number for the entire range of Reynolds number was accurate. As the Reynolds number was increased, the numerical solution tended to give a slightly higher value of the Strouhal number than did Matty's function. However, the Strouhal number generally decreased for increasing Reynolds number as indicated by Matty's study. 0.0
lO 3
lo 4
For ~ = 1/2, predicted values of the Strouhal number were lower than those given by Matty's function for low values of Reynolds number. However, as the Reynolds number increased the predic0.20 tions became more accurate. T he t r e n d of d e c r e a s i n g Strouhal number with increasing Reynolds number was also 0.15 supported by the prediction. The results were also similar for ;t = 1/4, though predicted 0.10 ~" Strouhal numbers slightly unm derestimated the empirical re• ,~=1 lation. Nonetheless, the trend ~ = 1/2 of decreasing Strouhal num0.05 • ;t = 1 / 4 ber with increasing Reynolds m number was again supported. I i I ]iilil f The variation in the Strouhal 0.00 104 105 number with)t for a fixed value of the Reynolds number was Reynolds Number also examined. Strouhal numF i g u r e 11 S t r o u h a l N u m b e r v e r s u s bers for 2 = 1/4 were generally Reynolds Number for a Flat greater than those for ;t = 1/2. P l a t e N e a r a Wall Matty's data indicate a simi-
2860 lar behavior between these values of A. However, the predicted Strouhal numbers at A = 1 were significantly higher than those at A = 1/2, though Matty's study indicates that these values should be nearly equal. The difference here m a y well liein the fact that Matty's study was for a 3D plate, while the predictions were for a 2-D plate.
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The predicted aerodynamic F i g m ~ 12 D ~ Coefficient versus forces changed with A, but Ground Clearance, A as expected for a bluffbody, there was little dependence of the drag on Reynolds number. Figure 12 presents the variation in drag coefficient with ;t. The trend for the drag to increase as ;t decreases is dearly apparent. At ;t = 0, the drag attained a value about 50% greater than t h a t obtained for a flat plate without a ground plane. This behavior was expected, for the increase in the drag on a body as it approaches a ground plane is well documented. The lift also increased as the ground clearance decreased, though the trend was not as prominent as for the drag. The predicted lift coefficient tended to decrease as Reynolds number increased. However, it must be noted that the magnitude o f the lift coefficient was quite small, and the lift may be considered to be i n s i ~ f i c a n t . 5.
CONCLUSIONS
This study proposed to demonstrate the ability to predict the phenomena of vortex shedding from 2-D bluffbodies. For every case tested with the exception of a fiat plate in contact with a ground plane, periodic vortex shedding was achieved. For the initial case involving a fiat plate in a uniform flow the results were favorable. Predicted Strouhal numbers agreed well with experimental data. Predicted pressures in the wake of the plate were much lower than the experimental data; however, the difference may be due to difficulties in experimentally measuring the pressures. Predictions of the drag also indicated deviations from the drag function provided by Roshko, but the accuracy of Roshko's function was unknown. With respect to parameters affecting the solution of the finite-difference equations, some significant results were discovered. Refinement of the finite-difference grid yielded only a slight improvement in the prediction of pressure distributions, and actually hampered the prediction of Strouhal numbers. The inclusion ofturbulence was found to have very little effect.
2861 The prediction of vortex shedding from a flat plate near a ground plane also yielded favorable results. The comparison of predicted Strouhal numbers with an empirically derived function demonstrated the ability to predict general trends in the shedding frequency. Quantitatively, there were significant differences in the prediction and the empirical function; however, this may be attributable to the fact that the prediction was 2-D while the function was derived for a 3-D case. Also, the expected increase in the aerodynamic force coefficients as the body approached the ground plane were accurately predicted. 6.
ACKNOWLEDGMENTS
The authors would like to thank Mr. Larry Socha and Mr. Jack Williams of Aerodynamics Department at Ford Motor Company for their help and support during the study. 7.
REFERENCES
1. Younis, B. Ain Lami (1988). On Modeling the Vortex Shedding From Bluff Bodies in Laminar and Turbulent Streams, Presented at the Seventh International Conference on Offshore Mechanics and Arctic Engineering, Houston, TX, February 7-i2, 1988. 2. Roshko, Anatol (1953). On the Development of Turbulent Wakes From Vortex Streets, NACA T e c h n i c a l N o t e 2913. 3. Roshko, Anatol (1954). A New Hodograph for Free-Streamline Theory, NACA T e c h n i c a l N o t e 316K 4. Roshko, Anatol (1954). On the Drag and Shedding Frequency of Two-Dimensional Bluff Bodies, NACA T e c h n i c a l Note 3169. 5. Matty, Rosemary R. (1979). V o r t e x S h e d d i n g F r o m S q u a r e P l a t e s N e a r a G r o u n d P l a n e : An E x p e r i m e n t a l S t u d y , M. S. Thesis, Texas Tech University. 6. Strickland, James H., Matty, Rosemary R., and Barton, Gregory H (1980); Vortex Shedding From Square Plates Perpendicular to a Ground Plane, AIAA J o u r n a l 18, pp. 715-716. 7. Patankar, Suhas V. (1980). N u m e r i c a l H e a t T r a n s f e r a n d F l u i d Flow. Hemisphere Publishing Corporation, New York, NY.