External Flow Around Submerged Bodies

Force a submerged object, the force acting on it determines the motion of the body. Two main surface interactions generate forces:

• Normal pressure $p$, which acts perpendicular to the surface. This is a result of the fluid compressing against the surface of the object.
• Shear stress ${\tau }_w$, which acts tangentially along the surface. This arises due to viscous effects as fluid layers slide past each other near the object.

The relative velocity $V_r$ is the speed of the fluid with respect to the object’s surface. It determines how pressure and shear forces develop as the fluid interacts with the object.

Integrating pressure and shear stresses over the object surface gives two forces:

• Drag force – in line with the relative velocity

$$F_D={\int }_S({\tau }_{w,x}-p_{s,x})dA$$

• Lift force – perpendicular to the relative velocity

$$F_S={\int }_S({\tau }_{w,y}-p_{s,y})dy$$

For ideal fluids (incompressible and inviscid), $F_D=F_L=0$.

For viscous fluids, if the object is symmetric and does not rotate, we have $F_D\ne 0$ and $F_L-0$.

If the object is asymmetric, then $F_D\ne 0$ and $F_L\ne 0$. A classic example of this is the airfoil, shown below.

Drag and Lift Coefficients

The distribution of pressure and shear stress over a submerged complex can be very complex. For practical analysis, we define dimensionless drag and lift coefficients.

Drag coefficient:

$$C_D=\frac{F_D}{\frac{1}{2}\rho V^{2}A}$$

which in turn lets us write drag force as

$$F_D=C_D\frac{1}{2}\rho V^{2}A$$

Lift coefficient:

$$C_L=\frac{F_L}{\frac{1}{2}\rho V^{2}A}$$

which lets us write lift force as

$$F_L=C_L\frac{1}{2}\rho V^{2}A$$