Page 88 - Computational Fluid Dynamics for Engineers

Problems 73

[10] Schubauer, G. B. and Skramstad: "Laminar boundary-layer oscillations and transition
on a flat plate". NACA Tech. Rept. 909, 1947.
[11] Hirsch, C : Numerical Computatian of Internal and External Flows, Volume 1, John
Wiley and Sons, N.Y., 1988.
[12] Cebeci, T.: An Engineering Approach to the Calculation of Aerodynamic Flows, Hori-
zons Pub., Long Beach, Calif, and Springer, Heidelberg, 1999.
[13] Rogers, S.E. and Kwak, D., "An Upwind Differencing Scheme for the Time Accurate
Incompressible Navier-Stokes Equations," AIAA J., Vol. 28, No. 2, pp. 253-262, 1990.

Problems

2-1. Multiplying the x-component of the momentum equation, Eq. (2.2.2). by
u, and using the definition of the substantial derivative, one obtains
<9cr
da xx da xy + (P2.1.1)
Dt\2 U ) gdx g xz uf x
dx dy dz
2 2
Adding the corresponding equations for D/Dt{l/2v ) and D/Dt(l/2w ), which
can be most easily derived by changing the variables in Eq. (P2.1.1) in cyclic
order, show that the resulting expression with V denoting the total velocity can
be written as
D l l d d d u
( ^r2\ ( P , P , P\ , (d(J Xx , da xy da xz
Dt \2 J g \ dx dy dzj g \ dx dy dz
v_ fda yx do yv da yz\ w_ (da zx da zy da zz
g \ dx dy dz J g \ dx dy dz

+ uf x + vf y + wf z (P2.1.2)

Equation (P2.1.2) is known as the kinetic energy equation. Its left-hand side
represents the rate of increase of kinetic energy per unit mass of the fluid as the
fluid moves along a streamline. The terms on the right-hand side, which can be
written in tensor notation as
V± (-^JL + d(Jj >J + f.
g \ dxi dxj
represent, respectively, the rates at which work is done on a unit mass of the
fluid by the pressure, by the viscous stresses, and by the body force per unit
/
mass, , to produce kinetic energy of the mean motion.

2-2. Noting that for incompressible flows the continuity and momentum equa-
tions in tensor notation can be written as
| ^ = 0 (P2.2.1)
dxj