The mathematical properties of the pressure-velocity and vorticity-velocity formulations of the equations of viscous flow are compared. It is shown that a vorticity-velocity formulation exists which has the interesting property that non-inertial effects only enter the problem through the implementation of initial and boundary conditions. This valuable characteristic, along with other advantages of the vorticity-velocity approach, are discussed in detail

Speziale, C. G.

English

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NASA Contr.lctor Report 178125 
lCASE REPORT NO. 86-35 
leASE 
ON THE ADVANTAGES OF THE VORTICITY-VELOCITY 
FORMULATION OF THE EQUATIONS OF FLUID DYNAMICS 
Charles G. Speziale 
Contract No. NASl-18l07 
May 1986 " 
FOR REFERENCE 
KOT TO liE Tutti FtlOM nus BOO" 
INSTITUTE FOR CO~~UTER APPLICATIONS IN SCIENCE AND ENGINEERING 
NASA Langley Research Center, Hampton, Virginia 23665 
Operated by the Universities Space Research Association 
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https://ntrs.nasa.gov/search.jsp?R=19860020629 2020-03-20T14:40:59+00:00Z
ON THE ADVANTAGES OF THE VORTICITY-VELOCITY FORMULATION 
OF THE EQUATIONS OF FLUID DYNAMICS 
Charles G. Speziale 
Institute for Computer Applications in Science 
NASA Langley Research Center, Hampton, VA 
and 
and Engineering 
23665-5225 
Georgia Institute of Technology, Atlanta, GA 30332 
Dedicated to Milton E. Rose 
on Occasion of his 60th Birthday 
Abstract 
The mathematical properties of the pressure-velocity and vorticity-
velocity formulations of the. equations of viscous flow are compared. It is 
shown that a vorticity-velocity formulation exists which has the interesting 
property that non-inertial effects only enter into the problem through the 
implementation of initial and boundary conditions. This valuable characteris-
tic, along with other advantages of the vorticity-velocity approach, are 
discussed in detail. 
Research was supported by the National Aeronautics and Space 
Administration under NASA Contract No. NASI-18107 while the author was in 
residence at ICASE, NASA Langley Research Center, Hampton, VA 23665-5225. 
i t\Ju, - 'OOIOI::it-
Two distinctly different approaches have been utilized in the literature 
for the numerical solution of the equations of viscous flow in three-
dimensions. In the more common approach, the momentum equation, which 
contains both the velocity and pressure, is solved numerically along with a 
derived Poisson equation for the pressure (Le., the pressure-velocity or 
primitive variable formulation [1-3]). The alternative approach is based on 
eliminating the pressure from the momentum equation by the application of the 
curl. In this manner, a vorticity transport equation is solved numerically in 
lieu of the momentum equation (Le., the vorticity-velocity formulation [4-
6] ) • The purpose of the present note is to explore in more detail the 
properties of these disparate numerical approaches. It will be shown that the 
vorticity-velocity formulation has a striking advantage when applied to 
problems in non-inertial frames of reference. More specifically, there exists 
an intrinsic vorticity-velocity formulation wherein all non-inertial effects 
(arising from both the rotation and translation of the frame of reference 
relative to an inertial framing) only enter into the solution of the problem 
through the implementation of initial and boundary conditions. This is in 
stark contrast to the pressure - velocity formulation where non-inertial 
effects appear directly in the momentum equation in the form of Coriolis and 
Eulerian accelerations--a state of affairs which can give rise to a variety of 
numerical problems [2]. A detailed exposition of this interesting property of 
the vorticity-velocity formulation will be presented along with a brief 
discussion of other advantages of this approach. 
For simplicity, we will restrict our attention to the analysis of viscous 
incompressible flow governed by the Navier-Stokes equation and continuity 
equation which, respectively, take the form 
a~ 
- + v • at 
-2-
v • v 
(1) 
0, (2) 
where ~ is the velocity vector, p is the pressure, and v is the kinematic 
viscosity of the fluid. Here, the validity of (1) requires that the external 
body forces be conservative and that the frame of reference be inertial. In 
an arbitrary non-inertial frame of reference (which can rotate with a time-
dependent angular velocity g(t) and translate with a time-dependent velocity 
~o(t) relative to its origin 0), the Navier-Stokes equation takes the more 
complex form [7] 
av 
. 
at + ~ • V ~ + n x r + n x (g x r) + ~o + 2n x ~ (3) 
Here, E is the position vector and the non-inertial terms on the left-hand 
side of (3) are, respectively, referred to as the Eulerian, centrifugal, 
translational, and Coriolis accelerations. The continuity equation still 
assumes the same form (2) in any non-inertial fram~ of reference. 
By the introduction of a modified pressure p which includes the 
centrifugal and translational acceleration potentials, the non-inertial form 
of the Navier-Stokes equation (3) can be simplified considerably. More 
specifically, (3) can be written in the equivalent form 
av a~ + ~ • V ~ + n x r + 2n x y = - VP + vV2 ~, (4) 
where 
-3-
. 
+ ~o • r. (5) 
In the pressure-velocity formulation, equation (4) is solved in conjunction 
with a Poisson equation for the pressure which is obtained by taking the 
divergence of (4)., Hence, the governing equations to be solved numerically in 
this approach can be summarized as follows: 
dV d~ + ~ • V ~ + n x £ + 2n x ~ = - VP + vV 2 ~, (6) 
- tr(y~ • ~) + 22 • ~, (7) 
subject to the initial and boundary conditions 
v 
,..., ~o ' at t (8) 
~ = ~: 1 on B. (9) 
In (7) and (9), tr(·) denotes the trace, ~ is the vorticity vector, and B 
denotes the boundary surface of the region. Of course, equations (6) and (7) 
must be solved subject to the continuity equation (2). Since we are 
considering general three-dimensional flow, a stream function solution does 
not exist. Hence, the solution for the velocity v 
,..., 
must be projected in some 
suitable fashion onto the space of solenoidal vectors. 
It is quite clear that the form of (6) and (7) (and, hence, their 
mathematical character) change depending on whether or not the frame of 
-4-
reference is inertial. Consequently, a particular numerical algorithm which 
may be optimal for a given class of flows in an inertial frame of reference 
may not be so for the same class of flows in a non-inertial framing. It will 
now be demonstrated that the vorticity-velocity formulation does not suffer 
from this deficiency. 
The vorticity-velocity formulation is based on the vorticity transport 
equation which is obtained by taking the curl of (4). This equation takes the 
form 
ale 
at + :!. • Vw (10) 
in any non-inertial frame of reference where 
v x V (11 ) 
is the vorticity vector. It is clear that the velocity and vorticity are also 
connected through the Poisson equation 
- 1 x le (12) 
which is a direct consequence of the vector identity 
v x (1 x ~) (13) 
The intrinsic vorticity E, defined by 
w w + 2g, (14) 
-5-
can be introduced which represents the vorticity relative to an inertial frame 
of reference. Since g is spatially homogeneous (Le., yg = 0), it is a 
simple matter to show that the non-inertial form of the vorticity-velocity 
formulation can be written as follows: , 
') 
aw 
+ ,),y2 
-+ v • vw w • Vv w 
at ~ 
(15) 
v
2 
v - V x w. (16) 
Equations (15) - (16) must be solved (in some region R with a boundary 
surface B) subject to the initial and boundary conditions 
w at t (17) 
on B. (18) 
Of course, it is well known that the vorticity, as well as the intrinsic 
vorticity, are solenoidal, i.e., 
v • W = 0, (19) 
and, hence, the solutions for and must, in some suitable fashion, be 
projected onto the space of solenoidal vectors. 
This vorticity-velocity formulation of fluid dynamics represented by 
equations (15) - (18) has the striking property that non-inertial effects only 
-6-
enter into the solution of the problem through the implementation of initial 
and boundary conditions. Consequently, the basic structure of the numerical 
algorithm (i.e., the numerical formulation of (15) - (16» will be independent 
of whether or not the frame of reference is inertial--a situation which 
greatly enhances the general applicability of any Navier-Stokes computer code 
which is developed based on this approach. 
At this point, a few comments should be made concerning the alternate 
ways in which the velocity field can be calculated in the vorticity-velocity 
formulation. Instead of solving the Poisson equation (16), it is possible to 
solve the defining equation for vorticity directly, i.e., 
v x ~ = w W - 22, (20) 
(see Gatski, Grosch, and Rose [6]). Of course, for plane or axisymmetric 
flows, there exists a stream function ~ such that [7] 
v = ~ x J,~ (21) 
v x (~ x J,~) W - 2g, (22) 
where ~ = J,X and X is the coordinate that the flow is independent of (for 
plane flows, (22) reduces to the Poisson equation 2 V ~ = W - 2n). While the 
motion of the frame of reference does enter into the equations of motion in 
these alternate vorticity-velocity formulations, it does so in a much less 
significant way than in the pressure-velocity formulation. To be specific, 
the transport equation which is solved (i.e., equation (15» does not contain 
'. 
-7-
any frame-dependent terms and, at each time step, the partial differential 
.,'; 
equation for the determination of the velocity field is only altered by the 
addition of a constant forcing function in the form of (the added term on 
the right-hand side of (20) and (22». 
Finally, it would be of value to mention some other advantages of the 
vorticity-velocity formulation. More difficulties have been known to arise in 
the implementation of pressure boundary conditions than vorticity boundary 
conditions [1,2] (of course, both boundary conditions must usually be 
derived). Difficulties in satisfying the continuity equation in the pressure-
velocity formulation have also been known to give rise to numerical 
instabilities [1]. Furthermore, in the vorticity-velocity approach, the 
vorticity vector is calculated directly. This is of considerable value since 
the vorticity field can play an important role in characterizing certain 
features of turbulence [8]. While it is certainly not being suggested that 
the pressure-velocity formulation be abandoned, this study does indicate that 
the vorticity-velocity formulation can have distinct advantages when applied 
to an important class of viscous flows. 
Acknowledgment 
The author would like to thank Dr. T. Gatski and Dr. M. Rose for some 
valuable comments and criticisms of the original draft of this paper. 
-8-
REFERENCES 
[1] A. J. CHORIN, Math. Comp., 22 (1968), 745. 
[2] G. P. WILLIAMS, J. Fluid Mech., 37 (1969), 727. 
[3] D. A. ANDERSON, J. C. TANNEHILL, and R. H. PLETCHER, "Computational Fluid 
Dynamics and Heat Transfer," McGraw-Hill, New York, 1984. 
[4] S. C. R. DENNIS, D. B. INGHAM, and R. N. COOK, J. Compo Phys., 33 (1979), 
325. 
[5] H. F. FASEL, "Numerical Solution of the Complete Navier-Stokes Equations 
for the Simulation of Unsteady Flows," Lecture Notes in Mathematics, No. 
771, Springer-Verlag, Berlin, 1980. 
[6] T. B. GATSKI, C. E. GROSCH, and M. E. ROSE, J. Camp. Phys., 48 (1982), 1. 
[7] G. K. BATCHELOR, "An Introduction to Fluid Dynamics," Cambridge 
University Press, London, 1967. 
[8] E. LEVICH and A. TSINOBER, Phys. Letters, 93A (1983), 293. 
Standard Bibliographic Page 
1. Report No. NASA CR-178125 12. Government Accession No. 3. Recipient's Catalog No. 
ICASE Report No. 86-35 
4. Title and Subtitle 5. Report Date 
ON THE ADVANTAGES OF THE VORTICITY-VELOCITY May 1986 
FORMULATION OF THE EQUATIONS OF FLUID DYNAMICS 6. Performing Organization Code 
7. Author(s) 8. Performing Organization Report No. 
Charles G. Speziale 86-35 
9. PfH?l~±re\t<t~~b¥ior;~tfiW{l eg4 AX«fi5Ti ca t ions 10. Work Unit No. in Science 
and Engineering 
11. C~fff!.. yr81mt No. Mail Stop 132C, NASA Langley Research Center 
Hampton, VA 23665-5225 
12. Sponsoring Agency Name and Address 13. Type of Report and Period Covered 
Contractor Report 
National Aeronautics and Space Administration 14. Sponsoring Agency Code 
Washington, D.C. 20546 505-31-83-01 
15. Supplementary Notes 
Langley Technical Monitor: Submitted to Applied Numerical 
J. C. South Mathematics 
Final Report 
16. Abstract 
The mathematical properties of the pressure-velocity and vorticity-
velocity formulations of the equations of viscous flow are compared. It is 
shown that a vorticity-velocity formulation exists which has the interesting 
property that non-inertial effects only enter into the problem through the 
implementation of initial and boundary conditions. This valuable 
characteristic, along with other advantages of the vorticity-velocity 
approach, are discussed in detail. 
" . 
. 
. 
17. Key Words (Sufgested by Authors(s» 
viscous f ow, vorticity-velocity 
18. Distr4bution Statement 3 - Fluid Mechanics and Heat 
formulation, Navier-Stokes equations Transfer 
59 - Mathematical and Computer 
Sciences (General) 
Unclassified - unlimited 
19. Security Classif.(of this report) 
Unclassified 120. Security Classif.(of this page) Unclassified 21. No. of Pages 122. Price 10 A02 
For sale by the National Technical Information Service, Springfield, Virginia 22161 
NASA-Langley, 1986 
End of Document 


On the advantages of the vorticity-velocity formulation of the equations of fluid dynamics

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