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M E 320 Professor John M. Cimbala Lecture 33
Today, we will :
- Begin discussion about Chapter 10 – Approximate solutions of the N-S equation
- Show how to nondimensionalize the equations of motion
- Discuss creeping flow (flow at very low Reynolds number)
VIII. APPROXIMATE SOLUTIONS OF THE NAVIER-STOKES EQUATION A. Introduction We have three ways to solve the differential equations of fluid flow:
- Analytically (Chapter 9) [solve exactly, but only for very simple problems]
- Numerically (Chapter 15) [use CFD on a computer to solve for thousands of cells]
- Approximately (Chapter 10) [ignore some terms in the N-S equation, then solve]
B. Nondimensionalization of the Equations of Motion
Now we substitute all of the above into Eq. 10-2 to obtain
Every additive term in the above equation has primary dimensions {m^1 L-2^ t -2^ }. To nondimensionalize the equation, we multiply every term by constant L /( ρ V^2 ), which has primary dimensions {m-1^ L^2 t 2 }, so that the dimensions cancel. After some rearrangement,
Thus, Eq. 10-5 can therefore be written as
Nondimensionalization vs. Normalization : Equation 10-6 above is nondimensional , but not necessarily normalized. What is the difference?
- Nondimensionalization concerns only the dimensions of the equation – we can use any value of scaling parameters L , V , etc., and we always end up with Eq. 10-6.
- Normalization is more restrictive than nondimensionalization. To normalize the equation, we must choose scaling parameters L , V , etc. that are appropriate for the flow being analyzed, such that all nondimensional variables ( t *, V *
G
, P *, etc.) in Eq. 10-6 are of order of magnitude unity. In other words, their minimum and maximum values are reasonably close to 1.0 (e.g., -6 < P *^ < 3, or 0 < P *^ < 11, but not 0 < P *^ < 0.001, or - < P *^ < 500). We express the normalization as follows: t *^ ~ 1, x G^ *^ ~ 1, V^ G^ *^ ~ 1, P *^ ~ 1, g G ^ ~ 1, ∇G~ 1
Navier-Stokes equation in nondimensional form:
Euler number, where 0 Eu (^2)
P P
ρ V
Inverse of Froude number squared,
where Fr
V
gL
Inverse of Reynolds number, where
Re
ρ VL
Strouhal number, where
St
fL V
If we have properly normalized the Navier-Stokes equation, we can compare the relative importance of various terms in the equation by comparing the relative magnitudes of the nondimensional parameters St, Eu, Fr, and Re.
