Numerical Solutions Applied To Heat Transfer And Fluid Mechanics Problems
This lecture is based upon the paper by B. P. Leonard, deriving the QUICK and QUICKEST methods. Note that we have already derived these methods via a different path in previous lectures. Take a look at Leonard’s Equation 23. Renaming the variables to match my notation, we have:
Notice that QUICK is one of those instances where you get something by thinking about practical bounds on the wave number in the stability analysis.
If you are having trouble with Equations 49 through 51,
remember that they work from the assumption that the density is a quadratic
function of x. For example, if I need to
know the density near the face between volumes i and
i+1, I work with the first three terms of a
To simplify notation Leonard introduced a new space variable . A positive sign of this new variable corresponds to the upstream direction. He’s avoiding use of in this context, since that denotes the length of the spatial mesh.
The two derivatives in the expression are approximated by finite difference, giving:
Assume that a spatial density distribution is advected without change in shape. The time history of density at the right face can be obtained using the density profile at the beginning of the time step upstream of the face. The relationship between time and space variables in these two ways of viewing density is:
So the time integral converts to a space integral as follows:
or in terms of the material Courant number c, we have
Consider a mass conservation equation that might result from multi-phase or reacting flow:
If the mass source term S(x,t) is small compared to the mass flux term then the Quickest difference formula is a reasonable approximation. However, in many such problems a quasi-equilibrium exists with:
Hence, the substitutions imbedded in Quickest:
are no longer valid. Use of a vanilla Quickest difference operator for flux terms does not give higher order accuracy. Be cautious of any numerical solution of flow equations that advertises both the Quickest difference method, and the ability to model multi-phase and/or reacting flows.