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On Vorticity Boundary Conditions

IP.com Disclosure Number: IPCOM000128296D
Original Publication Date: 1986-Dec-31
Included in the Prior Art Database: 2005-Sep-15
Document File: 6 page(s) / 23K

Publishing Venue

Software Patent Institute

Related People

Christopher R. Anderson: AUTHOR [+3]

Abstract

We discuss the problem of boundary conditions for the vortidty form of the 2-D Navier-Stokes equations and the Prandtl boundary layer equations. We present a new formu-lation of the vorticity boundary conditions and relate these conditions to those which are currently used in numerical algonthms. We present a finite difference scheme incorporating the new boundary conditions for the Prandtl boundary layer equations. Numerical results are presented.

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THIS DOCUMENT IS AN APPROXIMATE REPRESENTATION OF THE ORIGINAL.

On Vorticity Boundary Conditions

Christopher R. Anderson

Department of C -9 University of California at Los Angeles Los Angeles, California 90024

Number of pages: 18 Number of figures: 2 ANS classifii cations: 65,76

PartiaUy supported by INSF Mathematical Sciences Postdoctcral Research Fellowship, #NICS 83-11685 and ONR contract N00014-82-K-0335

ABSTRACT

We discuss the problem of boundary conditions for the vortidty form of the 2-D Navier-Stokes equations and the Prandtl boundary layer equations. We present a new formu-lation of the vorticity boundary conditions and relate these conditions to those which are currently used in numerical algonthms. We present a finite difference scheme incorporating the new boundary conditions for the Prandtl boundary layer equations. Numerical results are presented.

2. Results for the Prandtl Boundary Layer Equations

In this section we apply the ideas of the first section to the Prandtl boundary layer equa., tions. Miis set of equations, which we shall use to describe lamina flow over a half-infinite flat plate, is a simpler set of equations than the 2-D Navier Stokes equations, and yet when expressed in the vorticity formulation possesses the same problems with vorticity boundary conditions. We shall denve the continuous boundary conditions for the vorticity, and then implement a numencal method usmg these boundary conditions. Results of computations vnth this numerical method will be presented. A similar discussion concerning the implementation of numerical methods for the 2-D Navier Stokes equations will be presented in a forth coming paper (1].

In the vorticity form, the Prandd boundary equations are

(Equation Omitted)

with boundary conditions

(Equation Omitted)

Here (u,v) is the velocity and w the vorticity. The domain is the quarter plane

(Equation Omitted)

As is the case with the Navier Stokes equations, when one reconstructs the velocity field from the vorticity, it is not automatic that the bouridary condition on u in (2.5) will be satis-fied. Similarly, boundary conditions necessary to close equation (2.1) are not given. We pre-cede, as in the first section, to derive boundary conditions by expressing the condition on u in (2.5) as a

Stanford University Page 1 Dec 31, 1986

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On Vorticity Boundary Conditions

constraint on the vorticity and then setting time derivative of this constraint equal to zero. Using
(2.3), the constraint on the vorticity is

(Equation Omitted)

Thus we require

(Equation Omitted)

(2.8) is the desired boundary condition. We now consider a numerical method for solving these equations which incorporates these boundary conditions.

Our computational domain is the rectangular region described by the points (x, y) such that 0 :!-. x--. x. and 0 -~ y :s y.. The mesh we use is rectangular with widths dx and dy in the x and y direction respectively. The values of the vorticity and velocity are computed at...