Deck 14: Computational Fluid Dynamics

Question 1

Wind blowing over the cables of a bridge sheds alternating vortices resulting in oscillating forces acting on the cables. Which of the major parameters listed in Eq. 6.2.17 are the primary
Parameters which would be used in the description of this situation?
(A) Re and St
(B) Fr and St
(C) St and M
(D) Re and M

Answer 1

WIND BLOWING OVER THE CABLES OF A BRIDGE SHEDS ALTERNATING VORTICES RESULTING IN OSCILLATING
forces acting on the cables. Which of the major parameters listed in Eq. 6.2.17 are the primary
parameters which would be used in the description of this situation?
(A) Re and St
The shedding of the vortices would involve a shedding frequency leading to the
Strouhal number. The force of the wind on a cable would depend on the Reynolds
number. The flow would not be compressible so the Mach number would not enter.
Gravity would also not enter the problem so the Froude number would not enter.

Question 2

$\text {It is supposed that the power needed to propel a large dirigible depends on the speed \(V\),}$ $\text {diameter, \(D\), length \(L\), viscosity \(\mu\), density \(\rho\), and gravity \(g\). The relationship between these}$ $\text {variables can be expressed as:}$
(A) $\frac { P } { \rho V ^ { 2 } D ^ { 5 } } = f \left( \frac { D } { l } , \frac { v D } { V } \right)$
(B) $\frac { P } { \rho V ^ { 2 } D ^ { 5 } } = f \left( \frac { l } { D } , \frac { v } { V D } \right)$
(C) $\frac { P } { \rho V ^ { 2 } D ^ { 3 } } = f \left( \frac { l } { D } , \frac { v D } { V } \right)$
(D) $\frac { P } { \rho V ^ { 3 } D ^ { 2 } } = f \left( \frac { l } { D } , \frac { v } { V D } \right)$

Answer 2

D
$\frac { P } { \rho V ^ { 3 } D ^ { 2 } } = f \left( \frac { l } { D } , \frac { v } { V L } \right)$
$\text {The dimensions on the variables are}$
$$[ P ] = \frac { M L ^ { 2 } } { T ^ { 3 } } , \quad [ V ] = \frac { L } { T } , \quad [ D ] = L , \quad [ l ] = L , \quad [ v ] = \frac { L ^ { 2 } } { T } , \quad [ \rho ] = \frac { M } { L ^ { 3 } }$$
$\text {First, select the repeating variables, equal in number to the number of dimensions and}$ $\text {containing all dimensions in the problem. Let's select \(V , D\), and \(\rho\). Create the \(\pi\)-terms}$ $\text {by combining with each of the remaining variables, one at a time (cancel dimensions}$ $\text {as in Problem 1):}$
$$\pi _ { 1 } = \frac { P } { \rho V ^ { 3 } D ^ { 2 } } , \quad \pi _ { 2 } = \frac { l } { D } , \quad \pi _ { 3 } = \frac { v } { V L }$$
$\text {The \(\pi\)-terms are then related by}$
$$\frac { P } { \rho V ^ { 3 } D ^ { 2 } } = f \left( \frac { l } { D } , \frac { v } { V L } \right)$$

Question 3

$\text {The variables length \(l\), gravity \(g\), velocity \(V\), and density \(\rho\) can be combined into the}$ $\text {dimensionless term:}$
(A) $\rho V l ^ { 2 } / g$
(B) $\rho \mathrm { Vl } / \mathrm { g }$
(C) $V ^ { 2 } / l g$
(D) $V l ^ { 2 } / g$

Answer 3

C
$V ^ { 2 } / l g$
$\text {The dimensions on each of the variables are}$
$$[ l ] = L , \quad [ g ] = \frac { L } { T ^ { 2 } } , \quad [ V ] = \frac { L } { T } , \quad [ \rho ] = \frac { M } { L ^ { 3 } }$$
$\text {The density \(\rho\) is the only variable with \(M\) so it cannot enter the term since \(M\) is not }$$\text {able to be canceled out. Time \(T\) can be canceled by forming the ratio \(V ^ { 2 } / g\). That ratio}$$\text { leaves \(L\) in the numerator. If \(l\) is placed in the denominator, the term, called a \(\pi\)-term,}$$\text { is dimensionless:}$
$$\pi = \frac { V ^ { 2 } } { \lg }$$