5.3 Inviscid fluids

Figure 36: Uniform flow and development of a viscous boundary layer near a solid surface [3].

Eulerian fluid = perfect fluid

Shear stresses due to a viscous fluid may be neglected in many situations. Boundary layer (BL) analysis near the solid surfaces may be sufficient. Outside the BL, the fluid may be taken as inviscid. Constitutive equation (material) $$ \begin{equation} \boldsymbol{T} = -p \, \boldsymbol{1}, \quad T_{ij} = -p \, \delta_{ij} \tag{5.14} \end{equation} $$ where $ p=p(\rho,\theta) $, $ \rho(\boldsymbol{r},t) $ is the fluid density $ \theta(\boldsymbol{r},t) $ is the fluid temperature. Thermoelastic material as $ p = p(\rho,\theta) $

Incompressible Eulerian fluids

Compressibility may often be disregarded. Liquids are rarely considered compressible. Gases may also often be rendered incompressible ($ v < c/3 $). The pressure is no longer a state variable: $ p = p (\boldsymbol{r},t) $ and must be found from boundary conditions.

Linear momentum with Reynolds and Gauss $$ \begin{align} \frac{d}{dt} \int_V v_i \rho \, dV &= \int_V \partd{\rho v_i}{t} + \partd{}{x_k} (\rho v_i v_k) \, dV \tag{5.15}\\ &= \int_V \rho \partd{v_i}{t} + v_i \partd{\rho}{t} + v_i \partd{}{x_k} (\rho v_k) + \rho v_k \partd{v_i}{x_k} \tag{5.16}\\ &= \int_V \rho \underbrace{\left (\partd{v_i}{t} + v_k \partd{v_i}{x_k} \right)}_{\dot{v_i}} + v_i \underbrace{\left (\partd{\rho}{t} + \partd{}{x_k} (\rho v_k) \right )}_{=0 \Leftarrow\text{ mass conservation}} dV \tag{5.17}\\ &= \int_V \dot{v_i} \, \rho dV \tag{5.18} \end{align} $$ and consequently: $$ \begin{equation} \frac{d}{dt} \int_V \boldsymbol{v} \rho \, dV = \int_V \dot{\boldsymbol{v}} \rho \, dV \tag{5.19} \end{equation} $$

Equations of motion for Eulerian fluids. From Cauchy's equations of motion: $$ \begin{equation} \frac{d}{dt} \int_V \boldsymbol{v} \rho \, dV =\int_V \dot{\boldsymbol{v}} \rho \, dV = \int_V \nabla \cdot \boldsymbol{T} + \boldsymbol{b} \rho \; dV \tag{5.20} \end{equation} $$ General field equations: $$ \begin{equation} \dot{\boldsymbol{v}} = \partd{\boldsymbol{v}}{t} + (\boldsymbol{v} \cdot \nabla) \boldsymbol{v} = \frac{1}{\rho} \nabla \cdot \boldsymbol{T} + \boldsymbol{b} \tag{5.21} \end{equation} $$ The constitutive equation for a perfect fluid $ T_{ij} = -p \delta{ij} $ $$ \begin{equation} T_{ik,k} = - \partd{p \delta_{ik}}{x_k} = -\partd{p}{x_k} \delta_{ik} = -p_{,i} \tag{5.22} \end{equation} $$ The Euler equations $$ \begin{equation} \partd{\boldsymbol{v}}{t} + (\boldsymbol{v} \cdot \nabla) \boldsymbol{v} = -\frac{1}{\rho} \nabla p + \boldsymbol{b} \tag{5.23} \end{equation} $$

The governing equations for Eulerian fluids The Euler equations (momentum equations) $$ \begin{equation} \partd{\boldsymbol{v}}{t} + (\boldsymbol{v} \cdot \nabla) \boldsymbol{v} = -\frac{1}{\rho} \nabla p + \boldsymbol{b} \tag{5.24} \end{equation} $$

Conservation of mass (continuity): $$ \begin{equation} \partd{\rho}{t} + \nabla \cdot (\rho \boldsymbol{v}) = 0 \tag{5.25} \end{equation} $$

Together, (5.24) and (5.25) form 4 equations with 6 unknowns $ \boldsymbol{v}, p, \rho, \theta $. Thus in order to close equation system, an energy equation and a state equation $ p=p(\rho,\theta) $, are called for.

Equations of state

Ideal gas $ p =R\rho\theta $ Polytropic process $$ \begin{equation} p = p_0 \left ( \frac{\rho}{\rho_0} \right )^\alpha \tag{5.26} \end{equation} $$ where $ \alpha $ is constant and $ p_0 $ and $ \rho_0 $ are reference values.

Various processes

Isobaric process $ \Leftrightarrow $ constant pressure $ (\alpha=0) $
Isothermal process $ \Leftrightarrow $ constant temperature $ (\alpha=1) $
Isentropic process $ \Leftrightarrow $ constant entropy $ (\alpha=\kappa= c_p/c_\mu) $
Isochoric process $ \Leftrightarrow $ constant volume $ \nabla \cdot \boldsymbol{v} = 0 $ and $ (\alpha=\infty) $

5.3.1 Example 13: Sound waves

Audible sound consists of minute variations or perturbations of pressure which propagate as waves. One way to quantify sound, is to state the amount of pressure variation relative to atmospheric pressure. The threshold of hearing is generally reported as: $ p_h= 2\cdot 10^{-5} $ Pa, which is extremely small compared to the atmospheric pressure $ p_o = 1.01\cdot 10 ^5 $ Pa. The standard threshold of hearing can be stated in terms of pressure and the sound intensity in decibels can be expressed in terms of the sound pressure: $$ \begin{equation} I = 10 \log_{10} \left [ \frac{I}{I_0} \right ] = 10 \log_{10} \left [ \frac{p^2}{p_h^2} \right ] = 20 \log_{10} \left [ \frac{p}{p_h} \right ] \tag{5.27} \end{equation} $$

In the following we will show that sound waves propagate as elastic waves. The elastic waves correspond to small variations in pressure. We assume that sound propagation is an isentropic process governed by the Euler Eqs. (5.23), the mass conservation equation Eq. (5.8), and a constitutive equation $ p=p(\rho) $. A fluid is called barotropic if $ p=p(\rho) $ and $ \rho=\rho(p) $ are one-to-one relations. Further, the fluid is denoted an elastic fluid if it is both barotropic and inviscid. An elastic fluid is hyperelastic.

The atmospheric pressure $ p_0 $ and the $ \rho_0 $ represent together an equilibrium stationary state of the Euler Eqs. (5.23): $$ \begin{equation} 0 = -\frac{1}{\rho_0} \nabla p_0 = \boldsymbol{b} \tag{5.28} \end{equation} $$ Now, we introduce sound waves as perturbations in pressure ($ \tilde{p} $) and density ($ \tilde{\rho} $): $$ \begin{equation} \rho = \rho_0 + \tilde{\rho}, \qquad p= p_0 + \tilde{p} = p_0 + c^2 \, \tilde{\rho} \tag{5.29} \end{equation} $$ where the pressure perturbation was be eliminated by employing the constitutive equation to find: $$ \begin{equation} c^2 = \left . \frac{dp}{d\rho}\right |_{\rho=\rho_0} \tag{5.30} \end{equation} $$ and thus $ \tilde{p} = c^2 \, \tilde{\rho} $. Substitution of Eq. (5.29) into the linearized form of the Euler Eqs. (5.23) and subtraction of the stationary state in Eq. (5.28) yields the following system of equations: $$ \begin{align} \partd{\tilde{\rho}}{t} &= -\rho_0 \tag{5.31} \\ \partd{\boldsymbol{v}}{t} &= -\frac{1}{\rho_0} \nabla \tilde{p} = -\frac{c^2}{\rho_0} \nabla \tilde{\rho} \tag{5.32} \end{align} $$ The mass Eq. (5.31) may be differentiated with respect to time yield: $$ \begin{equation} \partdd{\tilde{\rho}}{t} = -\rho_0 \partd{}{t} \left ( \nabla \cdot \boldsymbol{v}\right ) = -\rho_0 \nabla \cdot \partd{\boldsymbol{v}}{t} = \rho_0 \nabla \cdot \frac{c^2}{\rho_0} \nabla \tilde{\rho} \tag{5.33} \end{equation} $$ where the latter equality follow from (5.32). The canonical linear wave equation results from (5.33): $$ \begin{equation} \partdd{\tilde{\rho}}{t} =c^2 \nabla^2 \tilde{\rho} \tag{5.34} \end{equation} $$ and we observe that $ c $ corresponds to the wave speed, which in this case is the velociy of sound, and may be found from the particular constitutive equation, e.g., Eq. (5.26).

For refence the speed of sound in air is c = 340 m/s, and in water it is c = 1460 m/s.