Page 301 - Intro to Tensor Calculus

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assigned to each variable are scales, of the appropriate dimension, which act as reference quantities which
reflect the order of magnitude changes expected of that variable over a certain range or area of interest
associated with the problem. An inappropriate magnitude selected for a characteristic constant can result
in a scaling where significant information concerning the problem can be lost. This is analogous to selecting
an inappropriate mesh size in a numerical method. The numerical method might give you an answer but
details of the answer might be lost.
In the above scaling of the variables occurring in the Navier-Stokes equations we let v 0 denote some
characteristic speed, p 0 a characteristic pressure, % 0 a characteristic density, L a characteristic length, g the
acceleration of gravity and τ a characteristic time (for example τ = L/v 0), then the barred variables v, p,
%,φ, t, x, y and z are dimensionless. Define the barred gradient operator by

∂ ∂ ∂
∇ = ˆ e 1 + ˆ e 2 + ˆ e 3
∂x ∂y ∂z
where all derivatives are with respect to the barred variables. The above change of variables reduces the
Navier-Stokes-Duhem equations

∂~v 2
% + %(~v ·∇)~v = −%∇φ −∇ p +(λ + µ )∇ (∇· ~v)+ µ ∇ ~v, (2.5.39)
∗
∗
∗
∂t
~ 2
% 0 v 0 ∂v % 0 v 0 p 0
to the form % + % ~ v · ∇ ~ v = −% 0g%∇ φ − ∇p
τ ∂t L L
(2.5.40)

∗
(λ + µ ) µ v 0 2
∗
∗
~
~
+ v 0 ∇ ∇· v + ∇ v.
L 2 L 2
2
Now if each term in the equation (2.5.40) is divided by the coefficient % 0 v /L, we obtain the equation
0
~
∂v −1 λ ∗ 1 1 2
~
~
~
~
S% + % v · ∇ v = %∇ φ − E∇p + +1 ∇ ∇· v + ∇ v (2.5.41)
∂t F µ ∗ R R
which has the dimensionless coefficients
p 0 % 0 V 0 L
E = 2 = Euler number R = = Reynolds number
% 0 v 0 µ ∗
v 0 2 L
F = = Froude number, g is acceleration of gravity S = = Strouhal number.
gL τv 0
Dropping the bars over the symbols, we write the dimensionless equation using the above coefficients.
The scaled equation is found to have the form

∂~v 1 λ ∗ 1 1 2
S% + %(~v ·∇)~v = − %∇φ − E∇p + +1 ∇ (∇· ~v)+ ∇ ~v (2.5.42)
∂t F µ ∗ R R