Two concepts to go further…#

Drag coefficient#

The ith component of the force on an obstacle can be written as

\[F_i = \iint_\Sigma \sigma_{ij} dS_j.\]

The total power injected in the flow is

\[P = {\boldsymbol{F}}\cdot {\boldsymbol{U}},\]

where \({\boldsymbol{U}}\) is the velocity relative to the object.

We define \(S\) as the surface projected on a cross-section perpendicular to the direction of the flow. Since the strength should scale as \(\rho U^2 S\), we define the drag coefficient (dimensionless) as:

\[C_D = \frac{2F_x}{\rho U^2 S}\]

The exact definition of the surface depends on the shape of the object (sphere, plate, etc.). The force can be due to pressure (force normal to the surface) or to friction (viscous force).

../_images/drag_coef_def.png

Fig. 10 Definition of the surface involved in the definition of the drag coefficient for different objects.#

For tubes or plates, an alternative quantity \(\lambda\) is used to quantify the drag. The pressure is measured at two different points and the difference of pressure is a direct measure of the friction:

\[\lambda = \frac{2}{\rho U^2 S} \frac{\Delta P D}{l},\]

where \(D\) is the diameter of the tube. Show that in the case of the tube, \(C_D = \lambda/4\).

../_images/drag_coef_def_pressure_drop.png

Fig. 11 The difference of pressure is measured by the difference of water height in two tubes.#

../_images/drag_coef_vs_Re_cylinder.png

Fig. 12 Evolution of the drag coefficient with the Reynolds number for a cylinder. The flow is stationary for small \(Re\). The wake becomes periodic for intermediate \(Re\).#

For the cylinder the drag becomes constant for very large Reynolds number. This implies that the energy dissipation scales like \(\rho U^3/D\) and does not depend on the viscosity.

Amplitude equations#

For some instabilities, it is possible to proposed a model describing some characteristics of the system close to the threshold of the instability without considering the fundamental equation of the fluid mechanics. This approach is called the amplitude equation and is based on the symmetries of the system.

One of the simplest examples is the case of the cylinder wake (Landau model, see p. 111 of GHP). We look for the equation describing the evolution of the complex amplitude of the perturbation \(A(t)\). We try

\[d_t |A|^2 = 2\sigma_r |A|^2 - 2B |A|^4,\]

which is equivalent to

\[d_t |A| = \sigma_r |A| - B |A|^3.\]

So a consistent equation for the complex amplitude can be

\[d_t A = \sigma A - B |A|^2 A.\]

From this equation, we find that \(|A_{eq}| = \sqrt{\sigma_r / B}\). Since we know that \(\sigma_r = 0\) for \(Re = Re_c\), we can use the Taylor expansion \(\sigma_r \propto (Re - Re_c)\), which would imply that \(|A_{eq}| \propto \sqrt{Re - Re_c}\). These relations are consistent with the measurements.

This model is also consistent with the observation that the characteristic time of the evolution of the perturbation scales like \(1/(Re - Re_c)\). This time diverges when the Reynolds number approaches the critical Reynolds number \(Re_c\). These characteristics are also observed in phase transitions, which can be described with the same Landau model.