Ch. 6—Differential Analysis of Fluid Flow

Main points:

  • Review of velocity and acceleration fields
  • Volumetric dilatation rate
  • Angular motion and deformation

Rotation of fluid particle given by

  • Differential form of continuity equation

for incompressible flow:
  • Stream function
  • Conservation of Linear Momentum

 

 

Inviscid flow

Euler equations—f = ma for a general fluid neglecting viscous effects (no shearing stresses). Valid for unsteady, compressible flow—can capture shocks. Widely used for design through numerical solutions.

If flow is steady and incompressible, can show from Euler equations that Bernoulli’s equation is valid along streamlines.

If flow is also irrotational, Bernoulli’s equation applies between ANY two points within that irrotational flow—no restriction to streamlines only.

If an incompressible, steady flow conserves mass and is also irrotational, the complex Euler equations can be reduced to Laplace’s equation in a velocity potential, f.

Similarly, for the same conditions but in 2D, the equations reduce to Laplace’s equation in the streamfunction:

and lines of constant streamfunction and constant potential are orthogonal!

Many useful inviscid flows can be created analytically by superimposing elemental solutions for source, sink, uniform flow, doublet. Such "practical" configurations include flow over a cylinder, flow over airfoil shapes.

Today, most inviscid flow solutions are obtained by solving the Euler or Laplace equations numerically by finite difference or finite volume methods.

Viscous flows: Effects of viscosity considered. Real flows are viscous. Results in no slip at solid boundaries, i.e., the velocity is zero at solid boundaries. The equations that govern motion (F=ma) are the Navier-Stokes equations. These are like the Euler equations except that the effects of shearing stresses are added. The stress terms we considered earlier are related to rates of strain. Some exact solutions exist for simple flows but today, numerical solutions to the N-S equations are fairly common.

 

 

 

Some notes on solving Laplace's equation numerically

 

 

Without viscous effects, the rotation of fluid particles cannot be changed leading to

 

This equation can be approximated by the finite-difference formulation:

When the grid spacing is equal in both directions, (), the equation simplifies to

 

Preferred method for solving is Gauss-Seidel iteration. The method works well for solving large systems of linear algebraic equations that have a certain level of "diagonal dominance." When the dominance is there the procedure is:

  1. Put the unknown having the coefficient largest in magnitude on the left-hand side of each equation.
  2. Make initial guesses for all unknowns.
  3. Using the most recently determined values on the right-hand side, compute new values for unknowns on the left-hand side. As you go through the list of equations, immediately update values so that you are always employing most recent values on the right.
  4. Pass through the list of equations repeatedly computing new values until changes are negligibly small.

Note: The required diagonal dominance is that the magnitude of the coefficient on the left is greater than or equal to the sum of the magnitudes of the coefficients to unknowns on the right with the greater than holding for one equation. If this procedure is working, each unknown takes a turn at appearing on the left so that as we pass through the list, each unknown is updated.