9.3. Drag-vorticity relation

Summary

  • Subject: Drag in terms of surface vorticity

  • Main conclusion: Drag coefficient of spherical particle can be expressed in terms of maximum surface vorticity.

  • Key idea The vorticity source is only at surface, and therefore the magnitude of drag is scaled by the maximum surface vorticity.

  • Reference:

Let us begin by the following equation of motion for incompressible inviscid flows.

(9.68)\[ \frac{\partial \mathbf{v}}{\partial t} + \mathbf{v} \cdot \nabla \mathbf{v} = - \frac{\nabla p}{\rho} + \mathbf{f} \]

Expanding \(\boldsymbol{\omega} \times \mathbf{v}\), we have

(9.69)\[\begin{split}\begin{split} \boldsymbol{\omega} \times \mathbf{v} \rightarrow \epsilon_{ijk} \omega_{j} v_{k} &= \epsilon_{ijk} \epsilon_{jmn} \frac{\partial v_{n}}{\partial x_{m}} v_{k} = ( \delta_{km} \delta_{in} - \delta_{kn} \delta_{im} ) \frac{\partial v_{n}}{\partial x_{m}} v_{k} = \frac{\partial v_{i}}{\partial x_{k}} v_{k} - \frac{\partial v_{k}}{\partial x_{i}} v_{k} \\ &= \frac{\partial v_{i}}{\partial x_{k}} v_{k} - \frac{\partial}{\partial x_{i}} \left( \frac{v^{2}}{2} \right) \end{split}\end{split}\]

The advection term can therefore be decomposed into two terms as

(9.70)\[ \mathbf{v} \cdot \nabla \mathbf{v} = \nabla \left( \frac{v^{2}}{2} \right) + \boldsymbol{\omega} \times \mathbf{v}\]

Eq. (9.68) becomes

(9.71)\[ \frac{\partial \mathbf{v}}{\partial t} = - \frac{1}{\rho} \nabla \left( p + \frac{\rho v^{2}}{2} \right) + \mathbf{v} \times \boldsymbol{\omega} + \mathbf{f} = - \frac{\nabla p_{T}}{\rho} + \mathbf{v} \times \boldsymbol{\omega} + \mathbf{f} \]

where \(p_{T} = p + \rho v^{2}/2\) is the total pressure. Integrating the equation of motion for the volume \(V\) enclosed by \(S\) yields

(9.72)\[ \frac{\partial }{\partial t} \iiint_{V} \rho \mathbf{v} dV = - \iint_{S} p_{T} \mathbf{n} dS + \iiint_{V} \left( \rho \mathbf{v} \times \boldsymbol{\omega} + \rho \mathbf{f} \right) dV\]

When a flow is steady the L.H.S. vanishes. In addition, taking \(S\), on which \(p_{T}\) is constant, shows that the external force must be balanced with the so-called vortex force \(\rho \mathbf{v} \times \boldsymbol{\omega}\) to maintain the steady flow [Saf95]. The drag force, \(\mathbf{F}\), acting on a body embedded in a incompressible inviscid flow can therefore be given by the total vortex force:

(9.73)\[ \mathbf{F} = \iiint_{V} \rho \mathbf{v} \times \boldsymbol{\omega} dV\]

If a flow is irrotational this relation gives no drag force. The proportionality inspired Legendre [Leg07] to establish a drag-vorticity relation for a body in a viscous fluid.

The Stokes drag for a spherical particle is given by

(9.74)\[ F_{D} = 6 \pi \mu u a\]

The velocity components are

(9.75)\[ v_{\theta} = u \sin \theta \left\{ 1 - \frac{a^{3}}{4 r^{3}} - \frac{3 a}{4 r} \right\}\]
(9.76)\[ v_{r} = - u \cos \theta \left\{ 1 - \frac{3 a}{2 r} + \frac{1}{2} \left( \frac{a}{r} \right)^{3} \right\}\]

The azimuthal component, \(\omega_{\varphi}\), of \(\boldsymbol{\omega}\) is only non-zero and is given by

(9.77)\[ \omega_{\varphi} = \frac{\partial v_{\theta}}{\partial r} + \frac{v_{\theta}}{r} - \frac{1}{r} \frac{\partial v_{r}}{\partial \theta} = \frac{3}{2} \left( \frac{a}{r^{2}} \right) u \sin \theta \]

At the solid surface,

(9.78)\[ \omega_{\varphi} = \frac{3 u \sin \theta}{2 a}~~~~\text{at}~r = a\]

The maximum vorticity at \(r = a\) is then

(9.79)\[ \omega_{\max} = \frac{3 u}{2 a}\]

The solid surface is the only source of the vorticity in the system. Hence, the magnitude of the drag would be scaled by \(\omega_{\max}\). Substituting Eq. (9.79) into Eq. (9.74) yields

(9.80)\[ F_{D} = 4 \pi \mu a^{2} \omega_{\max}\]

The drag coefficient is then

(9.81)\[ C_{D} = \frac{4 \pi \mu a^{2} \omega_{\max}}{\frac{1}{2} \rho u^{2} \pi a^{2}} = \frac{16 \mu}{2 \rho u a} \frac{\omega_{\max} a}{u} = \frac{16}{Re} \omega_{\max}^{*}\]

where

(9.82)\[ \omega_{\max}^{*} = \frac{\omega_{\max} a}{u}\]

For a bubble, the Hadamard-Rybczynski drag is given by

(9.83)\[ F_{D} = 4 \pi \mu u a\]

The velocity components are

(9.84)\[ v_{\theta} = u \sin \theta \left\{ 1 - \frac{a}{2 r} \right\}\]
(9.85)\[ v_{r} = - u \cos \theta \left\{ 1 - \frac{a}{r} \right\}\]

The azimuthal vorticity for this velocity fields is

(9.86)\[ \omega_{\varphi} = \frac{u a \sin \theta}{r^{2}}\]

Therefore,

(9.87)\[ \omega_{\varphi} = \frac{u \sin \theta}{a}~~~~\text{at}~r = a\]

The drag can be expressed in terms of this surface vorticity as

(9.88)\[ F_{D} = 4 \pi \mu a^{2} \omega_{\max}\]

and the drag coefficient is given by

(9.89)\[ C_{D} = \frac{16}{Re} \omega_{\max}^{*}\]

Therefore, the drag-vorticity relation for solid and fluid spheres in the Stoke regime can be integrated in the following form:

(9.90)\[ C_{D} Re = 16 \omega_{\max}^{*}\]

It is also shown that the Levich drag for the infinite \(Re\) limit has the same form as for the drag-vorticity relation. The drag is

(9.91)\[ F_{D} = 12 \pi \mu u a\]

As discussed in the previous section, the surface vorticity is given by

(9.92)\[ \omega_{\varphi} = \frac{3 u \sin \theta}{a}~~~~\text{at}~r = a\]

and

(9.93)\[ \omega_{\max} = \frac{3 u}{a}~~~~\text{and}~~~~\omega_{\max}^{*} = 3\]

For the drag and the maximum surface vorticity, Eq. (9.90) holds:

(9.94)\[ C_{D} Re = 16 \omega_{\max}^{*} = 48\]

Numerical simulations of spherical bubbles demonstrated that Eq. (9.90) is valid not only in the limiting cases of \(Re \ll 1\) and \(Re \rightarrow \infty\) but also at intermediate Reynolds numbers. The following drag correlation proposed by Mei et al. [MKL94] is applicable to a wide range of \(Re\):

(9.95)\[ C_{D} = \frac{16}{Re} \frac{16 + 3.315 \sqrt{Re} + 3 Re}{16 + 3.315 \sqrt{Re} + Re}\]

Substituting Eq. (9.95) into Eq. (9.90) yields the maximum surface vorticity expressed in terms of \(Re\):

(9.96)\[ \omega_{\max}^{*} = \frac{16 + 3.315 \sqrt{Re} + 3 Re}{16 + 3.315 \sqrt{Re} + Re}\]

It should be noted that Eq. (9.90) is no longer valid for solid spheres of \(Re > 1\). However, a linear relationship between \(C_{D} Re\) and \(\omega_{\max}^{*}\) can be found up to a certain \(Re\), below which a stable wake is formed.

Legendre [Leg07] also showed for ellipsoidal bubbles that

(9.97)\[C_{D} Re = 16 f(\chi, Re) \omega_{\text{max}}^{*}\]

where \(f(\chi, Re)\) is shape deformation factor: \(f(1, \infty) = 1\) and \(f(\chi, \infty)\) is given by Eq. (10.45). Note also that \(\omega_{\text{max}}^{*} = \chi\) at low \(Re\).