hw10

Chris Nkinthorn $\texttt{20190805}$

Prompt : (100 Points) Consider a 2-dimensional (2D) $[x, y]$, steady, incompressible, buoyant flow :

​ a. Write down the governing Navier-Stokes equations

​ b. Express the boundary layer equations in terms of Prandtl $\text{Pr}$, Reynolds $\text{Re}$, Grashof $\text{Gr}$, and Eckert $\text{Ec}$ numbers

​ c. Develop a corresponding set of boundary layer equations [momentum, energy], where the momentum $[\delta]$ and thermal thicknesses $\delta_T$ are not the same.

​ d. If Ec is small and if the convection terms are assumed of order the conduction terms, infer a relation for T as a function of other non-dimensional parameters.

Solution:

Part a: Write down the governing Navier-Stokes (N-S) equations, using the given assumptions for 2D $[x, y]$ flow: steady or constant with respect to time and incompressible or constant with respect to density. We include the buoyant term in the N-S equation as:

$$\rho[\vec{V} \bullet \nabla] \vec{V} = -\nabla p+\mu \nabla^{2} \vec{V}+ \overbrace{\rho g \beta \Delta T}^\texttt{Bouyant}.$$

Momentum conservation is dependent on three factors: the pressure, viscosity, and the aforementioned buoyancy. Buoyancy is a function of the gravitational acceleration, the local pressure, coupled with the Laplacian of the temperature distribution, per Fourier’s law.

Additionally to derive the boundary layer equation, the energy conservation expression is required. By the inclusion of specific heat capacity, the conservation for energy expression is seen to be a factor the material by the thermal conductivity, $k$, and viscous heat dissipation, $\Phi$, as:

$$\rho C_{p}[\vec{V} \bullet \nabla] T=k \nabla^{2} T+\Phi.$$

Part b: To express the boundary layer (BL) equations for momentum and energy in terms of Prandtl $\text{Pr}$, Reynolds $\text{Re}$, Grashof $\text{Gr}$, and Eckert $\text{Ec}$ numbers, we examine each BL individually. First, the momentum BL which depends on $\text{Gr}$ is premultiplied by $\frac{U_\infty^2}{L}$:

$$\rho[\vec{V}f \bullet \nabla] \vec{V}=-\nabla p+\mu \nabla^{2} \vec{V}+\rho g \beta \Delta T\\ \rho \frac{U_{\infty}^{2}}{L}[\vec{V} \bullet \nabla] \vec{V}=-\rho \frac{U_{\infty}^{2}}{L} \nabla p+\mu \frac{U_{\infty}}{L^{2}} \nabla^{2} \vec{V}+\rho g \beta \Delta T[\Delta T]_{r e f}\\ \boxed{ \rho[\vec{V} \bullet \nabla] \vec{V}=-\nabla p+\frac{1}{\operatorname{Re}} \nabla^{2} \vec{V}+\frac{G r}{\operatorname{Re}^{2}} }$$

Similarly, expressing the thermal energy boundary layer involves the same process of premultiplication on each side of the equation.

$$\rho C_{p}[\vec{V} \bullet \nabla] T=k \nabla^{2} T+\Phi\\ \rho C_{p} \frac{U_{\infty} \Delta T}{L}[\vec{V} \bullet \nabla] T=k \frac{\Delta T}{L^{2}} \nabla^{2} T+\frac{U_{\infty}^{2}}{L^{2}} \Phi\\ \rho \mathcal{C}_{p}[\vec{V} \bullet \nabla] T=\frac{k}{C_{p} \mu} \bullet \frac{\mu}{\rho U_{\infty} L} \nabla^{2} T+\left[\frac{\mu}{\rho U_{\infty} L}\right] \frac{U_{\infty}^{2}}{C_{p} \Delta T}\left[\frac{\Phi}{\mu}\right]\\ \boxed{ \rho C_{p}^{\prime}[\vec{V} \bullet \nabla] T=\frac{1}{\operatorname{Pr} \mathrm{Re}} \nabla^{2} T+\frac{E c}{\operatorname{Re}} \Phi }$$

Here, the coupling to the momentum boundary layer comes from the heat capacity of the fluid, the temperature and velocity distributions, as well as the Prandtl number. The Eckert number is found by definition as the ratio of advective transfer and the total possible heat dissipation.

Part c: To develop a corresponding set of boundary layer equations [momentum, energy], where the momentum $[\delta]$ and thermal thicknesses $[\delta_T]$ are not the same, order of magnitude analysis is employed. Let variation in the $x$ direction correspond to the free stream velocity $u$ normalized to $1$ and $\delta$ correspond to the high of the boundary layer on length scale much smaller than the free stream velocity.

$$\left[u u_{x}+v u_{y}\right]=-\frac{p_{x}}{\rho}+\frac{1}{\operatorname{Re}}\left[u_{x x}+u_{y y}\right]+\frac{G r}{\operatorname{Re}^{2}}\\ 1 \bullet \frac{1}{1}+\delta \bullet \frac{1}{\delta}=1+ \delta \overbrace{ \left[\frac{1}{1}+\frac{1}{\delta^{2}}\right] }^\text{simplify} +\delta^{2} G r$$

The reciprocal second positional derivative in the normal direction contributes much more than the component in the flow direction. This means that the expression simplifies:

$$\boxed{\left[u u_{x}+v u_{y}\right]=u_{e} u_{e x}+\frac{1}{\operatorname{Re}}\left[u_{y y}\right]+\frac{G r}{\operatorname{Re}^{2}}}$$

Similarly, analysis of the variation in the flow component normal to the boundary surface yields:

$$\left[u v_{x}+v v_{y}\right] = -\frac{p_{y}}{\rho}+v\left[v_{x x}+v_{y y}\right]\\ \overbrace{ 1 \bullet \frac{\delta}{1}+\delta \bullet \frac{\delta}{\delta} }^\text{LHS} = \frac{1}{\delta}+ \overbrace{ \delta^{2}\left[\frac{\delta}{1}+\frac{\delta}{\delta^{2}}\right] }^{\lll\text{LHS}} \\ \boxed{p_{y} \cong 0}$$

This leads to the conclusion that the pressure variation in the normal direction is very much negligible. At least in so far in comparison to the variation along the streamwise direction. Having investigated the vector components of momentum, additional order-of-magnitude analysis of the scalar energy expression yields:

$$\rho \mathcal{C}_{p}\left[u T_{x}+v T_{y}\right]=E c\left[u p_{x}+v p_{y}\right]+\frac{1}{\operatorname{Pr} \operatorname{Re}}\left[T_{x x}+T_{y y}\right]+\frac{E c}{\operatorname{Re}^{2}} \Phi\\ 1 \bullet \frac{1}{1}+\delta \bullet \frac{1}{\delta_{T}}=\frac{\delta^{2}}{\operatorname{Pr}} \overbrace{ \left[\frac{1}{1}+\frac{1}{\delta_{T}^{2}}\right] }^\text{simplify}$$

Again, the relative contribution of second derivatives is found and the streamwise direction neglected, so the simplification produces:

$$\boxed{ \rho C_{p}\left[u T_{x}+v T_{y}\right]=E c\left[u p_{x}+v p_{y}\right]+\frac{1}{\operatorname{Pr} \operatorname{Re}}\left[T_{y y}\right]+\frac{E c}{\operatorname{Re}^{2}}\left[u_{y}\right]^{2}.}$$

Part d: Additionally, if $\text{Ec}$ is small and the convection terms are assumed to be on the order of the conduction terms, to infer a relation for $\delta_{T}$ as a function of other non-dimensional parameters, we reexamine the relationship previously established, after order of magnitude simplification:

$$\rho \mathcal{C}_{p}\left[u T_{x}+v T_{y}\right]=E c\left[u p_{x}+v p_{y}\right]+\frac{1}{\operatorname{Pr} \operatorname{Re}}\left[T_{y y}\right]+\frac{E c}{\operatorname{Re}^{2}}\left[u_{y}\right]^{2}$$

so that application of ideal gas law in the far stream (noted with the subscript $\infty$) allows for correlation between the specific heat capacities for constant volume $c_v$ and constant pressure $c_p$. Algebraically, we can correlate with respect the ratio of these heat capacities, $\gamma$ .

$$\frac{p_{\infty}}{\rho_{\infty}}=R T_{\infty}=\left(c_{p}-c_{v}\right) T_{\infty}=\frac{\gamma-1}{\gamma} c_{p} T_{\infty}$$

Additionally, find the definition of the material parameter of the speed of sound of the fluid as:

$$c_{\infty}=\frac{\gamma p_{\infty}}{\rho_{\infty}}=(\gamma-1) c_{p} T_{\infty}$$

These, in conjunction with the order of magnitude analysis on the Eckert number which is the ratio of convective heat transfer to the possible diffusive transport. The term $\Delta T$, represents local deviation from the reference temperature, $\Delta T_{ref}$.

$$\begin{align} E c &=\frac{U_{\infty}^{2}}{c_{p}\left(\Delta T_{r e f}\right)}=\frac{U_{\infty}^{2}}{c_{p} T_{\infty}} \frac{T_{\infty}}{\left(\Delta T_{r e f}\right)}\\ & =(\gamma-1) \frac{U_{\infty}^{2}}{c_{\infty}^{2}} \frac{T_{\infty}}{\left(\Delta T_{r \sigma f}\right)}=(\gamma-1) M^{2} \frac{T_{\infty}}{\left(\Delta T_{r e f}\right)}\\ &\sim M^{2} \frac{T_{\infty}}{\left(\Delta T_{r e f}\right)} \sim 0.01 \frac{T_{\infty}}{\left(\Delta T_{r e f}\right)} \end{align}$$

Applying this to the form given in viscous boundary layer expression $\delta$ to the thermal BL $\delta_T$, produces:

$$1 \bullet \frac{1}{1}+\delta \bullet \frac{1}{\delta_{T}}=\frac{\delta^{2}}{\operatorname{Pr}}\left[\frac{1}{1}+\frac{1}{\delta_{T}^{2}}\right]\rightarrow \delta_{T} \cong \boxed{\frac{1}{\operatorname{Pr}^{1 / 2} \mathrm{Re}^{1 / 2}}} \\$$