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29.3.1 Computing Forces, Moments, and the Center of Pressure

The total force component along the specified force vector $\vec{a}$ on a wall zone is computed by summing the dot product of the pressure and viscous forces on each face with the specified force vector. The terms in this summation represent the pressure and viscous force components in the direction of the vector $\vec{a}$:


 \underbrace{F_a}_{\scriptstyle{total\ force\ component}} = \... ...{a} \cdot \vec{F_v}}_{\scriptstyle{viscous\ force\ component}} (29.3-1)


where      
  $\vec{a}$ = specified force vector
  $\vec{F_p}$ = pressure force vector
  $\vec{F_v}$ = viscous force vector
       

In addition to the actual pressure, viscous, and total forces, the associated force coefficients are also computed, using the reference values specified in the Reference Values panel (as described in Section  29.10). The force coefficient is defined as force divided by $\frac{1}{2}\rho v^2 A$, where $\rho$, $v$, and $A$ are the density, velocity, and area explicitly specified in the Reference Values panel. Finally, the net values of the pressure, viscous, and total forces and coefficients for all the selected wall zones are also computed.

The total moment vector about a specified center $A$ is computed by summing the cross products of the pressure and viscous force vectors for each face with the moment vector $\vec{r}_{AB}$, which is the vector from the specified moment center $A$ to the force origin $B$ (see Figure  29.3.1). The terms in this summation represent the pressure and viscous moment vectors:


 \underbrace{\vec{M}_A}_{\textstyle{total\ moment}} = \underb... ...e{\vec{r}_{AB} \times \vec{F_v}}_{\textstyle{viscous\ moment}} (29.3-2)


where      
  $A$ = specified moment center
  $B$ = force origin
  $\vec{r}_{AB}$ = moment vector
  $\vec{F_p}$ = pressure force vector
  $\vec{F_v}$ = viscous force vector
       

Figure 29.3.1: Moment About a Specified Moment Center
figure

Direction of the total moment vector follows the right hand rule for cross products.

In addition to the actual components of the pressure, viscous, and total moments, the moment coefficients are also computed, using the reference values specified in the Reference Values panel (as described in Section  29.10). The moment coefficient is defined as the moment divided by $\frac{1}{2}\rho v^2 A L$, where $\rho$, $v$, $A$, and $L$ are the density, velocity, area, and length explicitly specified in the Reference Values panel. Finally, the net values of the pressure, viscous, and total moments and coefficients for all the selected wall zones are also computed.

To reduce round-off error, a reference pressure (also specified in the Reference Values panel) is used to normalize the cell pressure for computation of the pressure force. For example, the net pressure force vector, acting on a wall zone, is computed as the vector sum of the individual force vectors for each cell face:


$\displaystyle \vec{F_p}$ $\textstyle =$ $\displaystyle \sum_{i=1}^{n}{ (p - p_{\rm ref}) A \hat{n}}$ (29.3-3)
  $\textstyle =$ $\displaystyle \sum_{i=1}^{n} p A \hat{n} - p_{\rm ref}\sum_{i=1}^{n} A \hat{n}$ (29.3-4)

where $n$ is the number of faces, $A$ is the area of the face, and $\hat{n}$ is the unit normal to the face.

The center of pressure is the average location of the pressure. The pressure varies around the surface of an object, such that $P = p(x)$. The general expression for determining the center of pressure is


 cp = \frac{\int x p(x) dx}{\int p(x) dx} (29.3-5)

However, the center of pressure of a wall is also defined as the point about which the moment on the wall(s) will be zero, that is, the point on the wall where all the forces balance. It is computed as follows:


$\displaystyle \vec{M}$ $\textstyle =$ $\displaystyle \vec{r} \times \vec{F}$ (29.3-6)
$\displaystyle \vec{M}$ $\textstyle =$ $\displaystyle 0$ (29.3-7)

where $\vec{F}$ is the force acting on the selected wall and $\vec{r}$ is the position vector from the center of pressure to the point where the force is applied.


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