Chapter 2: Differential Equations of Motion

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2 Chapter 2: Differential Equations of Motion

The need to understand the local detail of the flow drives the need for differential forms of the fluid flow equations. These differential forms are based on taking the limit of our previous conservation equations in the limit as the CV approaches an infinitesimal CV (or differential element). In general, these equations are derived from the evaluation of the conservation equations on an infinitesimal differential element level using vector calculus. More specifically, the approach gathers all the terms into a single volume integral that must apply everywhere, hence, provides a governing differential equation.

Example of infinitesimal control volume used to drive the differential form of the mass conservation equation.

Starting with the mass conservation equation, Eq (1) in Chapter 1, we initiate by moving the time-differential term into the volume intetegral. Also, applying divergence or Guass’ thoerom (which analytically relates volume and surface integrals), the second (flux) term can be expressed as a volume integral. The results from this procedure yields the following

(1) $\begin{equation*} \frac{\partial }{\partial t}\iiint_V{\rho dV} + \iint_S {\rho \vec{V}}\cdot\vec{ds}\rightarrow\iiint_V{\frac{\partial \rho }{\partial t}dV} + \iiint_V {\rho \nabla\cdot \vec{V}} dV=0. \end{equation*}$

The volume integral terms can then be combined to provide the following application of CV applied to a differential element:

(2) $\begin{equation*} \iiint_V{\left[\frac{\partial \rho }{\partial t} + \rho \nabla\cdot \vec{V}} \right]dV=0. \end{equation*}$

This equation must be satisfied throughout space and time and for any possible CV, which can only be achieved if the integrand also equals 0, hence, the conservation of mass in differential form given as

(3) $\begin{equation*} \frac{\partial \rho }{\partial t} + \rho \nabla\cdot \vec{V} =0 \end{equation*}$

The first term ( $\frac{\partial \rho }{\partial t}$ )can be described as the local time-rate-of-change of mass. The second term ( $\rho \nabla\cdot \vec{V}$ ) can be described as changes associated with mass moving into and away from the local point (regarded as advection).

A second approach to derive the conservation of mass in differential form can be developed by utilizing a Taylor-series expansion about the center of the infinitesimal element as indicated in Fig. 1. The mass flux into the left side can be given as $\left(\rho u\right) dy dz$ . The mass flux can then be extrapolated using the 1st term of the Taylor series to the right face as $\left(\rho u\right)=\left[\left(\rho u\right)+\frac{\partial \left(\rho u\right)}{\partial x} \Delta x\right] dy dz$ . The approach extends to all surfaces and yields the same governing equation, i.e. Eq. 3. See this video for the procedure:

The next aspect to consider is the limit as the flow is incompressible flow, i.e., $\rho$ is constant. This condition leads to a low that is Divergence Free. This literally implies that the conservation of volume and mass are the same. One can think of it as squishing in the x-direction is equally absorbed in the y and z directions. In this limit, conservation of mass is given as:

(4) $\begin{equation*} \nabla\cdot \vec{V} =0. \end{equation*}$

This simplified form drives analyses relevant to flow in liquids and low-speed applications, i.e., $M<0.3$ . Alternatively, high-speed flows and thermal flows that have variation in $\rho$ demand the conservative form given by Eq \label{eq:consMassDE2}.

Potential flow equations are a result of expressing Eq. (4) in terms of the velocity potential function, $\phi$ . As a general statement, $\vec{V}=\nabla\phi$ .

(5) $\begin{equation*} \nabla\cdot \vec{V} =\frac{\partial \vec{V}}{\partial x}+\frac{\partial \vec{V}}{\partial y}+\frac{\partial \vec{V}}{\partial z}=\frac{\partial^2 \phi }{\partial x^2}+\frac{\partial^2 \phi }{\partial y^2}+\frac{\partial^2 \phi }{\partial z^2}=\nabla^2\phi=0 \end{equation*}$

In the condition when the flow is irrotational ( $\nabla\times\vect{V}=0$ ) and incompressible ( $\nabla\cdot\vect{V}=0$ ), there is a connection with conservation of momentum through Bernoulli equation, Eq. 3 in Chapter 1, which links the fluid dynamics to provide pressures required for aerodynamic forces. Through these equations, combined with a variety of foundational, elementary flows, drive many aerodynamic theory and numerical predictions referred to as potential flow models. A nice tool and further discussion can be explored at potentialflow.com.

Example potential flow around a cylinder in cross flow.

Now move to the momentum conservation equation,

(6) $\begin{equation*} \frac{\partial }{\partial t}\iiint_V{\rho \vec{V} dV} + \iint_S {\rho \vec{V} \left(\vec{V}}\cdot\vec{ds}\right)=-\iint_s{pd\vec{s}}+\iint_s{\vec{\tau_w} ds}. \end{equation*}$

Similar to before, the time-differential term is moved within the volume integral. And again, the second (flux) term is expressed as a volume integral using the Divergence Theorem. Focusing on the LHS of Eq (6), the procedure implies:

(7) $\begin{equation*} \frac{\partial }{\partial t}\iiint_V{\rho \vec{V} dV} + \iint_S {\rho \vec{V} \left( \vec{V}\cdot\vec{ds} \right)}\rightarrow\iiint_V{\frac{\partial \rho \vec{V} }{\partial t} dV} + \iiint_V { \vec{V} \cdot \nabla \rho\vec{V}} dV \end{equation*}$

Now considering the RHS of Eq (6), and employing Gradient Theorom, we obtain:

(8) $\begin{equation*} -\iint_s{pd\vec{s}}+\iint_s{\vec{\tau_w} ds}\rightarrow-\iiint_V{\nabla p dV}+\iiint_V{\nabla \cdot \vec{\tau_w} dV} \end{equation*}$

The volume integrals can be combined to provide the LHS of a momentum CV assessment applied to a differential element:

(9) $\begin{equation*} \iiint_V{\left[\frac{\partial \rho \vec{V} }{\partial t} + \vec{V} \cdot \nabla \rho\vec{V}+ \nabla p - \nabla \cdot \vec{\tau_w} \right]} dV=0 \end{equation*}$

This equation must be satisfied throughout space and time and for any possible CV, which can only be achieved is the integrated also equals 0. This yields the conservation of mass in differential form given as:

(10) $\begin{equation*} \frac{\partial \rho \vec{V} }{\partial t} + \vec{V} \cdot \nabla \rho\vec{V}=-\nabla p + \nabla \cdot \vec{\tau_w} \end{equation*}$

When reduced in terms of incompressible flow with constant viscosity ( $\mu$ ), we obtain the Navier-Stokes equations which are given by

(11) $\begin{equation*} \rho\frac{\partial \vec{V} }{\partial t} + \rho\vec{V} \cdot \nabla \vec{V}=-\nabla p + \mu\nabla^2 \vec{V} \end{equation*}$

The solution of this set of equations, in combination with conservation of mass, is very difficult due to instabilities in the equations that lead to turbulence (see box below). A subset of these equations are the Euler equations, which are simply the form when $\mu\rightarrow 0$ .

Turbulence, by definition, are fluctuations in the flow with a turbulence cascade. This cascade involves larger structures, breaking into smaller and smaller structures that eventually lead to very high viscous mechanisms and losses. A physical description of turbulence is described here:

The characterization and modeling of these high frequency turbulent fluctuations drive the Reynolds Averaged Navier Stokes Equations (RANS) which are briefly described in this video:

Lastly, similar approaches are used to obtain the differential form of the energy equation. The result in terms of a low-speed flow is given as

(12) $\begin{equation*} \rho \frac{\partial e}{\partial t} + \rho\vec{V} \cdot \nabla e=\rho\dot{q}+k\nabla^2T-p\nabla\cdot\vec{V} + E_{visc}. \end{equation*}$

In general, the solutions to these equation sets drive fluid behavior and modeling associated with aerodynamics. These model equations serve to describe pressure, temperatures, and velocities throughout a fluid flow under a give set of both initial conditions (ICs) and boundary conditions (BCs). In steady aerodynamics we typically focus on BCs, which for a viscous calculation are:

Free-stream velocity
Free-steam pressure
No slip condition on solid (aerodynamic) surfaces

In Potential Flow (inviscid) models, our BCs are typically

Free-stream velocity
Free-steam pressure
Flow tangency or no penetration into solid (aerodynamic) surfaces

This flow tangency is a major consideration in the context of potential flow that drives much of the foundation of aerodynamics. And, in general, the goal of aerodynamics is to maximize lift while reducing drag. Therefore potential flow assumptions are often near valid and drive much of the most useful aerodynamic theory.

Computational Fluid Dynamics (CFD) is mostly driven by the Reynolds Averaged Navier Stokes Equations (RANS). RANS relies on averaging the velocities using a concept referred to as Reynolds decomposition. The decomposition describes time-varying velocities as the sum of the average and its perturbation from the average, i.e., $\vec{V}(t)=\overline{\vec{V}}+\vec{V}'(t)$ . The goal is to substitute this decomposed velocity, then average the resulting momentum equations over a turbulent time scale, $T$ . This procedure leads to simplifications and yields the RANS equations. RANS (or unsteady RANS, URANS) can be thought of as a solving for the mean velocity while estimating the fluctuations using a Turbulence Model.

The turbulence model intends to close out the Reynolds Stresses, which are terms that arise from averaging fluctuations. These terms can be reduced when utilizing the Boussinesq approximation which leads us to the form common to 1- and 2-equation turbulence models. The result is a familiar form given by:

(13) $\begin{equation*} \rho\frac{\partial \overline{\vec{V}} }{\partial t} + \rho\overline{\vec{V}} \cdot \nabla\overline{\vec{V}}=-\nabla p + \left(\mu+\mu_t\right)\nabla^2\overline{\vec{V}} \end{equation*}$

The main change is the added turbulence viscosity term, $\mu_t$ , which is an approximation of the Reynolds stresses common to the 1 and 2-equation turbulence models.

Example 1: Potential Flow Modeling of Airfoil (Boundary Element Method)

This is an example of a potential flow, boundary element method, applied to a cylinder.

Example 2: Potential Flow Model of Airfoil (XFOIL)

This is an example of the usage of Xfoil to analyze an airfoil

Example 3: CFD Model of Airfoil

This is an example of the application of StarCCM+ to an airfoil with a plain flap.

Key Skills: Consider these as you work through homework.

Know how to derive these equations using both calculation and differential methods.
Understand the basic concepts of derived models (potential flow and RANS) and what the methods mean and where they apply and do not apply.

2 Chapter 2: Differential Equations of Motion

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