CLASSIFICATION OF PARTIAL DIFFERENTIAL EQUATIONS (PDEs) IN CFD WITH EXAMPLE

Posted on Posted in Blog, Mechanical Engineering, Mid Level
Introduction

In CFD applications, computational schemes and specification of boundary conditions depend on the types of PARTIAL DIFFERENTIAL EQUATIONS. In many cases, the governing equations in fluids and heat transfer are of mixed types. For this reason, selection of computational schemes and methods to apply boundary conditions are important subjects in CFD.

Description

Partial differential equations (PDEs) in general, or the governing equations in fluid dynamics in particular, are classified into three categories:

(1) elliptic

(2) parabolic

(3) hyperbolic

 

  1. Elliptic Equations
  • A PDE is elliptic in a region if (B2 − 4AC < 0) at all points of the region.
  • An elliptic PDE has no real characteristics but only imaginary/complex characteristics.
  • A disturbance is propagated instantly in all directions within the region.
  • Examples of Elliptic PDEs are Laplace equation and Poisson equation.
  • The domain of solution for an elliptic PDE is a closed Region R.

CLASSIFICATION OF PARTIAL DIFFERENTIAL EQUATIONS IN CFD

  • Boundary value problem: Only boundary conditions are required to get the solution of elliptic equation.
  • Steady state temperature distribution of a insulated solid rod.
     2. Parabolic Equations
  • A PDE is parabolic in a region if (B2 − 4AC = 0) at all points of the region.
  • Time dependent problem: Example of parabolic PDEs is unsteady heat diffusion equation.

CLASSIFICATION OF PARTIAL DIFFERENTIAL EQUATIONS IN CFD

  • Marching type problem: The domain of solution for an parabolic PDE is an open Region.

CLASSIFICATION OF PARTIAL DIFFERENTIAL EQUATIONS IN CFD

  • Initial-Boundary value problems: Initial condition and two boundary conditions are required.
  • Examples: Boundary layers, jets, mixing layers, wakes, fully developed duct flows.
     3.  Hyperbolic Equations
  • A PDE is hyperbolic in a region if (B2 − 4AC> 0) at all points of the region.
  • Example of hyperbolic PDEs is wave equation.

CLASSIFICATION OF PARTIAL DIFFERENTIAL EQUATIONS IN CFD

  • The domain of solution for an parabolic PDE is an open Region.

CLASSIFICATION OF PARTIAL DIFFERENTIAL EQUATIONS IN CFD

  • Initial boundary value problem: Two Initial conditions and two boundary conditions are required.
  • Solution may be discontinuous (shock waves) : steady/unsteady compressible flows at supersonic speeds.
  • Method of Characteristics: A classical method to solve hyperbolic equations with two independent variables: Applicable to two-dimensional, steady, isentropic, adiabatic, irrotational flow of a perfect gas.
Physical Interpretation
  • Consider the flow of a body having velocity u in a quiescent fluid.
  • The movement of this body disturbs the fluid particles ahead of the body.
  • The propagation speed of disturbance would be equal to speed of sound, a.
  • The ratio of the speed of body to the speed of sound is called Mach number M=u/a.
  • Consider the steady two-dimensional velocity potential equation:

CLASSIFICATION OF PARTIAL DIFFERENTIAL EQUATIONS IN CFD

  • There are three types of PDEs for the three types of flows.
    1 Elliptic PDEs: Subsonic(M < 0).
    2 Parabolic PDEs: Sonic(M = 0).
    3 Hyperbolic PDEs: Supersonic(M > 0).

 

The physical situations these types of equations represent can be illustrated by the flow velocity relative to the speed of sound as shown in Figure 2.1.1. Consider that the flow velocity u is the velocity of a body moving in the quiescent fluid.The movement of this body disturbs the fluid particles ahead of the body, setting off the propagation velocity equal to the speed of sound a. The ratio of these two competing speeds is defined as Mach number, M=u/a.

For subsonic speed, M < 1, as time t increases, the body moves a distance, ut, which is always shorter than the distance at of the sound wave (Figure 2.1.1a). The sound wave reaches the observer, prior to the arrival of the body, thus warning the observer that an object is approaching. The zones outside and inside of the circles are known as the zone of silence and zone of action, respectively.

If, on the other hand, the body travels at the speed of sound, M = 1, then the observer does not hear the body approaching him prior to the arrival of the body, as these two actions are simultaneous (Figure 2.1.lb). All circles representing the distance traveled by the sound wave are tangent to the vertical line at the position of the observer. For supersonic speed, M > 1, the velocity of the body is faster than the speed of sound (Figure 2.1.1c). The line tangent to the circles of the speed of sound, known as a Mach wave, forms the boundary between the zones of silence (outside) and action (inside). Only after the body has passed by does the observer become aware of it.

 

CLASSIFICATION OF PARTIAL DIFFERENTIAL EQUATIONS IN CFD

 

The governing equations for subsonic flow, transonic flow, and supersonic flow are classified as elliptic, parabolic, and hyperbolic, respectively. We shall elaborate on these equations below. Most of the governing equations in fluid dynamics are second order partial differential equations. For generality, let us consider the partial differential equation of the form [Sneddon, 1957] in a two-dimensional domain.

CLASSIFICATION OF PARTIAL DIFFERENTIAL EQUATIONS IN CFD

Where the coefficients A, B, C, D, E, and F are constants or may be functions of both independent and/or dependent variables. To assure the continuity of the first derivative of u, ux = u/∂x and uy=∂u/∂y. We write

CLASSIFICATION OF PARTIAL DIFFERENTIAL EQUATIONS IN CFD

Since it is possible to have discontinuities in the second order derivatives of the dependent variable along the characteristics, these derivatives are indeterminate.

CLASSIFICATION OF PARTIAL DIFFERENTIAL EQUATIONS IN CFD

This happens when the determinant of the coefficient matrix in (2.1.3) is equal to zero.

CLASSIFICATION OF PARTIAL DIFFERENTIAL EQUATIONS IN CFD

Depending on the value of B2 – 4AC, characteristic curves can be real or imaginary.

For problems in which real characteristics exist, a disturbance propagates only over a finite region (Figure 2.1.2). The downstream region affected by this disturbance at point A is called the zone of influence. A signal at point A will be felt only if it originates from a finite region called the zone of dependence of point A.

The second order PDE is classified according to the sign of the expression ( B2 – 4AC).

(a) Elliptic if B2 – 4AC < 0

In this case, the characteristics do not exist.

(b) Parabolic if B2 – 4AC = 0

In this case, one set of characteristics exists.

(c) Hyperbolic if B2 – 4AC > 0

In this case, two sets of characteristics exist.

 

Note that (2.1.1) resembles the general expression of a conic section,

AX2 + BXY+ CY2 + DX+ EY+ F = 0                                 (2.1.8)

in which one can identify the following geometrical properties:

B2 – 4AC < 0 ellipse

B2 – 4AC = 0 parabola

B2– 4AC > 0 hyperbola

This is the origin of terms used for classification of partial differential equations.

CLASSIFICATION OF PARTIAL DIFFERENTIAL EQUATIONS IN CFD

 

References
  1. COMPUTATIONAL FLUID DYNAMICS by T. J. CHUNG, University of Alabama in Huntsville.
  2. Lecture notes on Computational Fluid Dynamics by Dr. Tariq Talha, College of EME, NUST, Islamabad, Pakistan.

 

Magoosh

Leave a Reply