13. Incompressible Fluid Mechanics

Chapter of a introductory course in incompressible fluid mechanics:

• statics

• kinematics

• governing equations

• non-dimensional equations

• vorticity dynamics

• low-\(Re\) exact solutions

• high-\(Re\) flows, incompressible inviscid irrotational flows:

  • vorticity dynamics and Bernoulli theorems

  • aeronautical applications

• boundary layer

• instability and turbulence

13.1. Navier-Stokes Equations

The kinematic constraints (link to Non-dimensional Equations of Fluid Mechanics?)

\[\nabla \cdot \vec{v} = 0\]

replaces mass balance in the governing equation and implies \(\frac{D \rho}{D t} = 0\), i.e. all the material particles have constant density in time.

If …

(13.1)\[\begin{split}\begin{cases} \rho \left[ \frac{\partial \vec{u}}{\partial t} + \vec{u} \cdot \nabla ) \vec{u} \right] - \mu \nabla^2 \vec{u} + \nabla P = \rho \vec{g} \\ \nabla \cdot \vec{u} = 0 \end{cases}\end{split}\]

with the proper initial conditions, boundary conditions and - if required - compatibility conditions.

Compatibility condition

A compatibility condition is needed if the velocity field is prescribed on the whole boundary \(\partial V\) of the domain \(V\),

\[\vec{u}\bigg|_{\partial V} = \vec{b}_n \ .\]

The compatibility condition reads

\[\oint_{\partial V} \vec{b} \cdot \hat{n} = 0 \ ,\]

to ensure that the conudary conditions are consistent with the incompressibility constraint, as it is readily proved using divergence theorem on the velocity field in \(V\),

\[0 \equiv \int_V \underbrace{\nabla \cdot \vec{u}}_{= 0} = \oint_{\partial V} \hat{v} \cdot \hat{n} = \oint_{\partial V} \vec{b} \cdot \hat{n} \ .\]

13.2. Vorticity

A dynamical equation for vorticity \(\vec{\omega} := \nabla \times \vec{u}\) reailty follows taking the curl of Navier-Stokes equations (13.1)

(13.2)\[\frac{D \vec{\omega}}{D t} = (\vec{\omega} \cdot \nabla) \vec{u} + \nu \Delta \vec{\omega} \ ,\]

i.e. vorticity can be stretched-tilted by the term \((\vec{\omega} \cdot \nabla) \vec{u}\), or diffused by the term \(\nu \Delta \vec{\omega}\).

…

13.3. Bernoulli theorems

For an incompressible fluid, the advective term \((\vec{u} \cdot \nabla) \cdot \vec{u}\) can be recasted as

\[(\vec{u} \cdot \nabla) \cdot \vec{u} = \vec{\omega} \times \vec{u} + \nabla \frac{|\vec{u}|^2}{2} \ ,\]

so that the momentum equation in Navier-Stokes equations (13.1) for fluids with uniform density \(\rho\) reads

(13.3)\[ \rho \left[ \frac{\partial \vec{u}}{\partial t} + \vec{\omega} \times \vec{u} + \nabla \frac{|\vec{u}|^2}{2} \right] - \mu \Delta \vec{u} + \nabla P = \rho \vec{g} \ .\]

Starting from the form (13.3), different forms of Bernoulli theorems are readilty derived with the proper assumptions.

Theorem 13.1 (Bernoulli theorem along path and vortex lines in steady flows)

In a steady incompressible inviscid flow with conservative volume forces, \(\vec{g} = - \nabla \chi\), the Bernoulli polynomial is constant along path (everywhere tangent to the velocity field, \(\hat{t}(\vec{r}) \parallel \vec{u}(\vec{r})\)) and vortex lines (everywhere tangent to the vorticity field, \(\hat{t}(\vec{r}) \parallel \vec{\omega}(\vec{r})\)), i.e. the directional derivative of the Bernoulli polynomial in the direction of the velocity or the vorticity field is identically zero,

\[\hat{t} \cdot \nabla \left( \frac{|\vec{u}|^2}{2} + \frac{P}{\rho} + \chi \right) = 0 \ .\]

The proof readily follows taking the scalar product with a unit-norm vector \(\hat{t}\) parallel to the local velocity or vorticity, and noting that \(\hat{t} \cdot \vec{u} \times \vec{\omega}\) is zero if either \(\hat{t} \parallel \vec{v}\) or \(\hat{t} \parallel \vec{\omega}\).

Theorem 13.2 (Bernoulli theorem in irrotational inviscid steady flows)

In a steady incompressible inviscid irrotational flow with conservative volume forces, \(\vec{g} = - \nabla \chi\), the Bernoulli polynomial is uniform in the whole domain, since its gradient is identically zero

\[\nabla \left( \frac{|\vec{u}|^2}{2} + \frac{P}{\rho} + \chi \right) = 0 \qquad \rightarrow \qquad \frac{|\vec{u}|^2}{2} + \frac{P}{\rho} + \chi = 0 \ .\]

Theorem 13.3 (Bernoulli theorem in irrotational inviscid flows)

In an incompressible inviscid irrotational flow with conservative volume forces, \(\vec{g} = - \nabla \chi\), the Bernoulli polynomial is uniform in the connected irrotational regions of the domain - but not constant in time in general - , since its gradient is identically zero

\[\nabla \left( \frac{\partial \phi}{\partial t} + \frac{|\vec{u}|^2}{2} + \frac{P}{\rho} + \chi \right) = 0 \qquad \rightarrow \qquad \frac{\partial \phi}{\partial t} + \frac{|\vec{u}|^2}{2} + \frac{P}{\rho} + \chi = C(t) \ .\]

being \(\phi\) the velocity potential used to write the irrotational velocity field as the gradient of a scalar function \(\vec{u} = \nabla \phi\).

Note

The assumption of inviscid flow is not directly required if irrotationality holds. Anyway the inviscid flow assumption may be required to make irrotationality condition holds. Lookinig at the vorticity equation (13.2) the assumption of negligible viscosity prevents diffusion of vorticity from rotational regions to irrotational regions.

Note

A barotropic fluid is defined as a fluid where the pressure is a function of density only, \(P(\rho)\). For this kind of flows it’s possible to find a function \(\Pi\) so that

\[d \Pi = \frac{d P}{\rho} \ .\]

The results of this section derived for a uniform density flow hold for a barotropic fluid as well, replacing \(\frac{P}{\rho}\) with \(\Pi\).