Progress in validating incompressible Navier-Stokes codes is described using a predictor/corrector scheme. The flow field under study is the impulsive start of a circular cylinder and the unsteady evolution of the separation bubble. In the current code, a uniform asymptotic expansion is used as an initial condition in order to correctly capture the initial growth of the vortex sheet. Volocity fields at selected instants of time are decomposed into vectorial representations of Navier-Stokes equations which are then used to analyze dominant contributions in the boundary-layer region

Davis, Sanford S.

English

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NASA Technical Memorandum 101 037 = 
Computational Studies of 
an Impulsively Started 
Viscous Flow 
Sanford S. Davis 
(bASA-IB-101C37) CCMEGTAZICIAI SIUCIES Ci H8Y-2  1190 
7 P  CSCL 20D 
A b  X l ! i E U L S l V E Z L  E1161Ef V l S C C t f  E L C Y  (HASA) 
Unclah  
G3/3U 019 1704 
December 1988 
NASA 
National Aeronautics and 
Space Administration 
https://ntrs.nasa.gov/search.jsp?R=19890011819 2020-03-20T02:46:58+00:00Z
NASA Technical Memorandum 101 037 
Computational Studies of 
an Impulsively Started 
Viscous Flow 
Sanford S. Davis, Ames Research Center, Moffett Field, California 
December 1988 
NASA 
National Aeronautics and 
Space Administration 
Ames Research Center 
Moffett Field, California 94035 
COMPUTATIONAL STUDIES OF AN IMPULSIVELY STARTED VISCOUS FLOW 
Sanford S. Davis 
Chief, Fluid Dynamics Research Branch, NASA Ames Research Center, Moffett Field, 
California 
ABSTRACT Progress in validating incompressible Navier-Stokes codes is described 
using a predictor/corrector scheme. The flow field under study is the impulsive start of a 
circular cylinder and the unsteady evolution of the separation bubble. In the current code, 
a uniform asymptotic expansion is used as an initial condition in order to correctly capture 
the initial growth of the vortex sheet. Velocity fields at selected instants of time are 
decomposed into vectorial representations of Navier-Stokes equations which are then 
used to analyze dominant contributions in the boundary-layer region. 
1. Introduction 
interior of a viscous fluid by examining flows about simple geometries such as cylinders. With 
respect to the prediction of viscous drag, cylinder flows are closely related to airfoils (except 
for a broader wake, higher drag, and more severe adverse pressure gradients). This can be an 
advantage in fundamental studies because the large lateral length scales allow more details to be 
measured (or computed). In addition, the transient viscous flow about a cylinder will be of 
great help in understanding the physical processes that govern the development of separation 
and drag on practical configurations and their ultimate minimization and control. 
One approach to the Navier-Stokes equations is to examine the short-time behavior due 
to impulsive starts. Such studies are reported in Wang [ 11 and [2], Collins and Dennis [3], 
Dennis and Staniforth [4], Bar-Lev and Yang [5 ] ,  and Ta Pho Loc [6] .  These early-time solu- 
tions, based as they are on asymptotic approximations, tend to diverge at moderate times. The 
flow at longer times must be dominated by large-scale vortex structures and its early time 
behavior does not seem to be important, at least in the global sense. The fully developed flow 
seems to be quite independent of the initial conditions, considering the broad general agreement 
among the previous investigations. 
The connection between the short- and long-time solutions is the onset of a wake insta- 
bility that converts the initially symmemc flow into an asymmetric vortex shedding mode. The 
exact mechanism that transforms a symmetric boundary-value problem into an unsteady wake- 
dominated flow is still unclear, although the trigger mechanism is believed to be unavoidable, 
but small, free-stream disturbances. 
The objective of this paper is to combine an analytic short-time solution with a predic- 
tor-corrector algorithm to march the unsteady Navier-Stokes equations forward in time. Unlike 
previous solutions, a uniformly valid initial flow satisfying the Navier-Stokes equations is used 
to start the calculation. The algorithm is very simple, using an implicit Euler forward integra- 
tion scheme with an upwind biased, third-order approximation for the convection term. The 
details of the solution, along with some experimental data, are presented in ref. [7]. 
started circular cylinder up to a nondimensional time 0 of about 7, based on the free-stream 
velocity and radius. Computations are presented for three combinations of grid spacing and 
outer boundary location. A detailed analysis of the force and acceleration vectors contributing 
to the Navier-Stokes equations is presented in graphical form. In this way, the relative impor- 
tance of the unsteady, convective, pressure-gradient, and viscous terms is made apparent. The 
onset of instability, and the prediction and analysis of the dominant instability mode will be the 
subject of a subsequent paper. 
Many attempts have been made to understand the complex physical processes in the 
The plan of the paper is to frrst compute the symmetric flow field of an impulsively 
1 
The most important result of the vector analysis shows that the flow near the surface of 
the body varies slowly in time, but the inviscid flow field away from the body has a large 
dynamic component. This dynamic component is due to the unsteady boundary-layer growth 
and could have a significant bearing on the subsequent wake instability. 
2. Computational Results 
In this section computational results are given and compared with other investigations. 
Pressure and vorticity on the surface are found to be consistent with previous calculations, and 
the inclusion of realistic initial conditions is shown to eliminate a "ringing" effect. 
Static pressure distributions for the three grids are shown in Figs. 1 and 2. The pres- 
sure distribution in Fig. 1 at T = 0.1 is compared with an analytical representation given by 
Wang in ref. [2]. The static pressure is a very sensitive parameter and is determined by three 
competing effects-the curvature of the wall normal velocity that has a cosine distribution that 
strongly affects both the forward and rear stagnation points; the convective acceleration that is 
symmetric fore and aft; and the unsteady term that has both symmemc and unsymmemc com- 
ponents. The three computed solutions are similar, except for a small discrepancy at the two 
stagnation points. The analytical solution shows the same trend, although the pressure falls to 
a deeper trough. The difference, however, between the minimum pressure and the rear 
stagnation pressure compares quite well. (Another analytical solution for the short-time impul- 
sive pressure is given in ref. [ 5 ] ,  but this solution was not reconcilable with Wang's formula.) 
Figure! 2 shows excellent agreement among the computed solutions and the solutions given by 
Dennis and Staniforth [4] at time T = 1.0. 
Turning to the time evolution of the flow, Fig. 3 shows the variation of forward stag- 
nation pressure. Figure 3 shows results for three computational grids. The sensitivity of the 
outer boundary is apparent for T > 3.0 where there is a slight, but consistent discrepancy. 
Other investigations show the stagnation pressure to fall to about 1.10 at infinitely long times, 
but this is an unrealistic result because a wake instability sets in at about T = 14. Also shown 
in Fig. 3 is the trend from Wang's asymptotic small-time solution where the stagnation pres- 
sure drops very quickly when T = O( 1). Figure 4 shows a reference calculation using the 
inviscid pressure distribution superimposed on the present computations. At the initial instant 
of time, the value of Cp is 1 .O and jumps quickly to a very high value. After about two 
cycles, the oscillation damps out and approaches the previous results with a slight residual 
oscillation. 
Since the kinematics of the flow field seem to be adequately represented by the compu- 
tational procedure, it is worthwhile to look a little deeper into the equations themselves. At 
each point in space, the Navier-Stokes equations can be represented by a vector that is in equi- 
librium among its four components: convection, diffusion, time derivative, and pressure gra- 
dient. The relative importance of these effects is presented in Fig. 5 at T = 0.14. The left 
figure shows solution vectors at 12 points on a circle at r = 1.1 within the highly viscous 
flow regime very soon after the initiation of motion. These vectors indicate a strong balance 
between diffusion and unsteady effects in the direction parallel to the body and between con- 
vection and pressure gradient in the normal direction. This verifies the essential uncoupling of 
the viscous, rotational flow field (the former) and the inviscid, irrotational flow (the latter). 
The right figure shows the instantaneous boundary layer flow profiles at T = 0.14 which are 
before the onset of flow separation. 
3. Concluding Remarks 
Physically plausible initial conditions were used to demonstrate that the evolving 
unsteady flow quickly forgets its early history. Individual field quantities are very useful, but 
they will not replace an analysis of the full dynamics of the flow field. A first step toward such 
a dynamical analysis was presented by examining the closure of the vector components of the 
full Navier-Stokes equation during the initial instant of motion. 
2 
References 
[ 13 Wang C-Y: J. Math and Phys. (1967). 
[2] Wang C-Y: J. Applied Mech. 34 (1967). 
[3] Collins W. M. & Dennis, S. C. R.: J. Fluid Mech. 64. (1973). 
[4] Dennis S. C. R. & Staniforth A. N.: Roc. 2nd Intl. Conf. on Numerical Methods in 
Fluid Dynamics (1971). 15-19 Sept. 1970, Berkeley, CA. 
[5] Bar-Lev M. & Yang H. T.: J. Fluid Mech. 12 (1975). 
[6] Ta Pho Loc: J. Fluid Mech. (1980). 
[7] Davis, S., Cho, Y-C, & McLachlan, B.: Proc. 1st Natl. Fluid Dynamics Congress 
(1988), 25-28 July 1988, Cincinnati, OM. 
-3 1 I 1 I 1 
0 45 90 130 180 
0 ,  deg 
0 
-.5 
N -1.0 
3 
9 
-1.5 
a, -2.0 
v) 
1 
n 
11 
Cl 
0 -2.5 
-3.0 
-3.5 
0 DENNIS AND STANIFORTH (1971) 
0 CURRENT CODE 
45 90 130 180 
8. deg 
Fig. 1. Static pressure distribution on the 
surface of an impulsively started circular 
cylinder at T = 0.10, Re = 100: open 
circles, mesh A; open squares, mesh B; 
open triangles, mesh C; closed circles, 
analytical solutions, Wang [ 11, [21. Staniforth [4]. T= 1.0. 
Fig. 2. Static pressure distribution on the 
surface of an impulsively started circular 
cylinder. Re = 100: open circles, 
meshes A, B, and C; closed circles, 
numerical computation of Dennis and 
3 
I 1 1 I I 
2 4 6 8 0 2 4 6 8 0 .8 ' 
T = tUia T = tU/a 
Fig. 3. Time history of the stagnation pres- 
sure on an impulsively started circular cylin- 
der. Re = 100: solid, mesh A; dash, 
mesh B; dots, mesh C; heavy line, analytical 
solution, Wang [ 11, [2]. 
Fig. 4. Time history of the stagnation pres- 
sure on an impulsively started circular cyl- 
inder. Re = 100: solid, mesh A; dash, 
mesh B; dots, mesh C; heavy line, solution 
without uniformly valid pressure as initial 
condition. 
V ------ 
I 
I 
I b. 
C .  
p ........... 472 VIr 
I 
I 
I 
Fig. 5. Flow field vector decomposition near the body at r = 1.1, Re = 100. Left: vector 
equations; right: instantaneous velocities. T = 0.14, scale factor = 1.0. 
4 
1. Report No. 
NASA TM- 10 103 7 
7. AuthorM 
2. Government Accession No. 
Sanford S .  Davis 
19. Security Classif. (of this report) 20. Security Classif. (of this pagel 21. No. of pages 
Unclassified Unclassified 7 
9. Performing Organization Name and Address 
22. Price 
A02 
Ames Research Center 
Moffett Field, CA 94035 
12. Sponsoring Agency Name and Address 
National Aeronautics and Space Administration 
Washington, DC 20546-000 1 
3. Recipient's Catalog No. 
5. Report Date 
December 1988 
6. Performing Organization Code 
8. Performing Organization Report No. 
A-8 82 95 
10. Work Unit No. 
505-60-3 1 
11. Contract or Grant No. 
13. Type of Report and Period Covered 
Technical Memorandum 
14. Sponsoring Agency Code 
15. Supplementary Notes 
Point of Contact: Sanford S .  Davis, Ames Research Center, MS 260-1, Moffett Field, CA 94035 
(4 15) 6 9 4 4  197 or FTS 4 6 4 4  197 
Paper to be presented at Fourth Asian Congress of Fluid Mechanics, August 19-23, 1989, in Hong Kong. 
16. Abstract 
Progress in validating incompressible Navier-Stokes codes is described using a predictor/corrector 
scheme. The flow field under study is the impulsive start of a circular cylinder and the unsteady evolu- 
tion of the separation bubble. In the current code, a uniform asymptotic expansion is used as an initial 
condition in order to correctly capture the initial growth of the vortex sheet. Velocity fields at selected 
instants of time are decomposed into vectorial representations of Navier-Stokes equations which are 
then used to  analyze dominant contributions in the boundary-layer region. 
17. Key Words (Suggested by Author(s1) I 18. Distribution Statement 
Viscous flows 
Cylinders 
Impulsive start 
Navier-S tokes 
Unclassified - Unlimited 
Subject Category - 34 


Computational studies of an impulsively started viscous flow

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