Hypersonic and High-Temperature Gas Dynamics Chapter 6 Notes

Chapter 6 is titled "Viscous Flow: Basic Aspects, Boundary Layer Results, and Aerodynamic Heating"
==Introduction==
==Governing Equations for Viscous Flow: Navier-Stokes Equations==
==Similarity Parameters and Boundary Conditions==
(Missed first 35 minutes of class, 7/30/2013)
It is useful to nondimensionalize the Navier-Stokes equations. We will consider a 2-D steady flow and ignore the normal stresses (τxx and τyy). We will use the following nondimensional variables:
:\bar\rho = \frac{\rho}{\rho_\infty} \qquad \bar u = \frac{u}{V_\infty} \qquad \bar v = \frac{v}{V_\infty} \qquad \bar p = \frac{p}{p_\infty}
:\bar e= \frac{e}{c_v T\infty} \qquad \bar\mu = \frac{\mu}{\mu\infty} \qquad \bar x = \frac{x}{c} \qquad \bar y = \frac{y}{c} 
:\bar k = \frac{k}{k_\infty} \qquad \bar T = \frac{T}{T_\infty}
where the subscript ∞ represents reference values and c is a characteristic length.
With these substitutions, the Navier-Stokes equations for 2-D, steady flow (neglecting volumetric heating) become:
Continuity
:\frac{\partial (\bar\rho \bar u)}{\partial \bar x} + \frac{\partial (\bar\rho \bar v)}{\partial \bar y} = 0
X-Momentum
:\bar\rho \bar u \frac{\partial \bar u}{\partial \bar x} + \bar\rho \bar v \frac{\partial \bar u}{\partial \bar y} = -\frac{1}{\gamma M_\infty^2} \frac{\partial \bar p}{\partial \bar x} + \frac{1}{Re_\infty}\frac{\partial}{\partial \bar y}\left[ \bar\mu \left(\frac{\partial\bar v}{\partial \bar x} + \frac{\partial \bar u}{\partial \bar y} \right) \right] 
Y-Momentum
:\bar\rho \bar u \frac{\partial \bar v}{\partial \bar x} + \bar\rho \bar v \frac{\partial \bar v}{\partial \bar y} = -\frac{1}{\gamma M_\infty^2} \frac{\partial \bar p}{\partial \bar y} + \frac{1}{Re_\infty}\frac{\partial}{\partial \bar x}\left[ \bar\mu \left(\frac{\partial\bar v}{\partial \bar x} + \frac{\partial \bar u}{\partial \bar y} \right) \right] 
Energy
: \begin{align} \bar\rho \bar u \frac{\partial \bar e}{\partial \bar x} + \bar\rho \bar v \frac{\partial \bar e}{\partial \bar y} = &-\frac{\gamma (\gamma - 1)}{2}M_\infty^2 \left[ \bar\rho \bar u \frac{\partial}{\partial \bar x}(\bar u^2 + \bar v^2) + \bar\rho \bar v \frac{\partial}{\partial \bar y}(\bar u^2 + \bar v^2) \right] 
\\ \\
&+ \frac{\gamma}{Pr_\infty Re_\infty} \left[ \frac{\partial}{\partial \bar x} \left( \bar k \frac{\partial \bar T}{\partial \bar x} \right) + \frac{\partial}{\partial \bar y} \left( \bar k \frac{\partial \bar T}{\partial \bar y} \right) \right] 
\\ \\
&- (\gamma - 1)\left[ \frac{\partial (\bar u \bar p)}{\partial \bar x} + \frac{\partial (\bar v \bar p)}{\partial \bar y} \right] 
\\ \\ 
&+ \gamma(\gamma - 1) \frac{M_\infty^2}{Re_\infty} \left( \frac{\partial}{\partial \bar x} \left[ \bar \mu \bar v \left( \frac{\partial \bar v}{\partial \bar x} + \frac{\partial \bar u}{\partial \bar y} \right) \right] + \frac{\partial}{\partial \bar y} \left[ \bar \mu \bar u \left( \frac{\partial\bar v}{\partial \bar x} + \frac{\partial \bar u}{\partial \bar y} \right) \right] \right) \end{align}
(missed some discussion here, 7/30/2013)
For air at STP, Pr = 0.71. Like μ, k, cp, it is only a function of temperature for non-chemically-reacting flows.
===Wall Boundary Conditions===
For a viscous, continuum flow, friction prevents the flow from slipping at the wall. This is the "no-slip" boundary condition at the wall. At the wall,
:u = v = 0
If there is mass transfer at the wall (i.e. transpiration cooling or ablation cooling), the wall boundary condition is
:u = 0 \qquad v = v_w \quad \text{the specified normal wall velocity}
If the wall is at a constant temperature,
:T = T_w \quad \text{the specified wall temperature}
For variable wall temperature along the surface,
:T = T_w(s) \quad \text{where s represents the surface position}
The heat transfer boundary condition is
:q_w = -k\left( \frac{\partial T}{\partial n} \right) _w 
where \left( \frac{\partial T}{\partial n} \right) _w is the normal temperature gradient in the gas, at the wall.
For an adiabatic wall boundary condition,
:\left( \frac{\partial T}{\partial n} \right) _w = 0
The wall temperature that results from the boundary condition is defined as the "adiabatic wall" temperature, Taw, sometimes called the "equilibrium" temperature.
For high-speed viscous flows, either a specified, uniform wall temperature or an adiabatic wall is usually assumed. Note that the heat transfer boundary condition requires a coupled solution with heat conduction in the surface material.
(missed a little bit before break, 7/30/2013)
Now, we will apply some appropriate simplifying assumptions for boundary layers.
==Boundary-Layer Equations for Hypersonic Flow==
Boundary layer equations are obtained by making suitable order-of-magnitude reductions to the Navier-Stokes equations.
Consider 2-D steady flow. The non-dimensional form of the x-momentum equation is:
:\bar\rho \bar u \frac{\partial \bar u}{\partial \bar x} + \bar\rho \bar v \frac{\partial \bar u}{\partial \bar y} = -\frac{1}{\gamma M_\infty^2} \frac{\partial \bar p}{\partial \bar x} + \frac{1}{Re_\infty}\frac{\partial}{\partial \bar x}\left[ \bar\mu \left(\frac{\partial\bar v}{\partial \bar x} + \frac{\partial \bar u}{\partial \bar y} \right) \right] 
Consider the boundary layer along a long, flat plate of length c. The boundary layer in this high speed flow is very thin in comparison to the length.
: \delta \ll c
where δ is the boundary layer thickness.
The continuity equation is:
:\frac{\partial (\bar\rho \bar u)}{\partial \bar x} + \frac{\partial (\bar\rho \bar v)}{\partial \bar y} = 0
:\bar u = 0 \quad \text{at the wall}
:\bar u = 1 \quad \text{at the edge of the boundary layer}
So \bar u is O[1]. Similarly, \bar x and \bar p are O[1]. However, \bar y varies from 0 to δ/c and δ<<c, then \bar y is O[δ/c]. If we assume c = 1, then \bar y is O[δ].
Performing an order-of-magnitude analysis on the continuity equation gives:
:\frac{[O(1)][O(1)]}{O(1)} + \frac{[O(1)][\bar v]}{O(\delta)} = 0
Hence \bar v is O[δ].
Performing an order-of-magnitude analysis on the x-momentum equation gives:
: O(1) + O(1) = -\frac{1}{\gamma M_\infty^2}O(1) + \frac{1}{Re_\infty} \left[ O(1) + O\left( \frac{1}{\delta ^2} \right) \right]
Another boundary layer theory assumption is that the Reynolds number is large enough that
: \frac{1}{Re_\infty} = O[\delta ^2]
Now the previous equation becomes:
: O(1) + O(1) = -\frac{1}{\gamma M_\infty^2}O(1) + O(\delta ^2) \left[ O(1) + O\left( \frac{1}{\delta ^2} \right) \right]
The O[δ2]O[1] term is much smaller than the rest of the terms, so it will be neglected. The remaining terms of the x-momentum equation are:
:\bar\rho \bar u \frac{\partial \bar u}{\partial \bar x} + \bar\rho \bar v \frac{\partial \bar u}{\partial \bar y} = -\frac{1}{\gamma M_\infty^2} \frac{\partial \bar p}{\partial \bar x} + \frac{1}{Re_\infty}\frac{\partial}{\partial \bar y}\left( \bar\mu \frac{\partial \bar u}{\partial \bar y} \right) 
or in dimensional form,
:\rho u \frac{\partial u}{\partial x} + \rho v \frac{\partial u}{\partial y} = -\frac{\partial p}{\partial x} + \frac{\partial}{\partial y}\left( \mu \frac{\partial u}{\partial y} \right) 
This equation holds for boundary layer flows with large Reynolds number, such that Eqn. 6.19 holds.
The y-momentum for 2-D, steady flow is
:\bar\rho \bar u \frac{\partial \bar v}{\partial \bar x} + \bar\rho \bar v \frac{\partial \bar v}{\partial \bar y} = -\frac{1}{\gamma M_\infty^2} \frac{\partial \bar p}{\partial \bar y} + \frac{1}{Re_\infty}\frac{\partial}{\partial \bar x}\left[ \bar\mu \left(\frac{\partial\bar v}{\partial \bar x} + \frac{\partial \bar u}{\partial \bar y} \right) \right] 
Performing an order-of-magnitude analysis on the y-momentum equation gives:
: O(\delta) + O(\delta) = -\frac{1}{\gamma M_\infty^2}\frac{\partial \bar p}{\partial \bar y} + O(\delta^2) \left[ O(\delta) + O\left( \frac{1}{\delta} \right) \right]
This analysis shows that:
:\frac{\partial \bar p}{\partial \bar y} = O(\delta)
or smaller, assuming γM∞2 = O(1). Because δ is very small, \frac{\partial \bar p}{\partial \bar y} must be very small, therefore,
:\frac{\partial p}{\partial y} = 0
This states that at a given x location in the boundary layer, the pressure is constant through the boundary layer normal to the surface. Hence,
: p = p(x) = p_e(x)
throughout the boundary layer, where the subscript e denotes the boundary layer edge pressure, which can be determined from inviscid flow calculations.
For large hypersonic Mach numbers, M∞2 in Eqn. 6.24 is very large, then
:\frac{\partial\bar p}{\partial\bar y} \neq 0
and does not need to be small. Therefore, for hypersonic flow, the assumption that p is constant normal to the surface in the boundary layer is not always valid. This does not preclude the normal pressure gradient from being zero. For large hypersonic Mach numbers, then the pressure gradient term...
(missed the last 3 boards, 7/30/2013)
===Summary of Boundary Layer Equations===
Assumptions:
*δ << c
*Re = O[1/δ2]
*Equations are for 2-D, steady, viscous flow
The Navier-Stokes equations for the boundary layer become:
Continuity
:\frac{\partial (\rho u)}{\partial x} + \frac{\partial (\rho v)}{\partial y} = 0
X-Momentum
:\rho u \frac{\partial u}{\partial x} + \rho v \frac{\partial u}{\partial y} = \frac{\text{d} p_e}{\text{d} x} + \frac{\partial}{\partial y}\left[ \mu \frac{\partial u}{\partial y} \right] 
Y-Momentum
:\frac{\partial p}{\partial y} = 0
Energy
:\rho u \frac{\partial h}{\partial x} + \rho v \frac{\partial h}{\partial y} = \frac{\partial}{\partial y} /left( k \frac{\partial T}{\partial y} /right) + u \frac{\text{d} p_e}{\text{d} x} + \mu \left( \frac{\partial u}{\partial y} \right) ^2
This set of equations for boundary layer flow is non-linear, but it is easier to solve than the equation set without boundary layer assumptions. We need to close the boundary layer equation set with an equation of state and a constitutive relation.
#Perfect gas equation of statep = \rho RT
#Calorically perfect gash = c_p T
The boundary layer equations have the following boundary conditions.
*At the wall,y = 0 \qquad u = 0 \qquad v = 0 \qquad T = T_wor for an adiabatic wall\left( \frac{\partial T}{\partial n} \right)_w = 0
*At the boundary layer edgey \rightarrow \infty \qquad u \rightarrow u_e \qquad T = T_e
The boundary layer thickness, δ, is not known ahead of time so the edge boundary layer boundary condition is given at large y. The boundary layer equations were created for subsonic and supersonic lows, and they can be applied to hypersonic flows.
Note that \frac{\partial \bar p}{\partial \bar e} \neq 0 does not hold for all hypersonic flows. If M∞ is high enough for viscous dissipation to create temperatures high enough to cause chemical reactions within the boundary layer, then diffusion of chemical species and energy changes resulting from chemical reactions must be included.
Note that the application of Eqns. 6.27-6.32, pg. 276, to moderate hypersonic conditions still yields useful results.
==Hypersonic Boundary-Layer Theory: Self-Similar Solutions==
The similarity solutions in this section are applicable to subsonic, supersonic, and hypersonic conditions. However, for the hypersonic case, viscous dissipation will lead to high-temperature, chemically-reacting flows. In this course, we will highlight only the effects of viscosity and thermal conductivity in combustion with high Mach numbers.
Consider the boundary layer shown in Fig 6.5, pg. 278. The left side shows the boundary layer velocity profile at two different axial locations drawn in a physical plane, u(x1,y) ≠ u(x2,y).
Under the appropriate independent variable transformation from (x,y) to (ξ,η), the profile is independent of axial position along the surface (shown on the right side of Fig 6.5, pg. 278). Thus, in the transformed coordinates, u = u(η), independent of ξ. Boundary layers that exhibit this behavior are self-similar boundary layers and have self-similar solutions. These solutions for incompressible flows were originally obtained by Blasius in 1908.
We will transform the boundary layer equations (Eqns. 6.27-6.32, pg. 276) from (x,y) space to (ξ,η) space using:
:\xi = \int_0^x \rho_e u_e \mu_e \text{d}x
:\eta = \frac{u_e}{\sqrt{2 \xi}} \int_0^y \rho \text{d}y
where ρe, ue, and μe are functions of x only; therefore, ξ = ξ(x).
This transform is called the Lees-Dorodnitsyn transformation. We will use the following steps in this transformation.
#Transform the independent variables. The chain rule of calculus is used to transform the differential terms. Details are provided on pg. 280-281. The x-momentum equation in (ξ,η) space is:\frac{\partial \psi}{\partial\eta} \left[ \rho_e u_e \mu_e \frac{\partial u}{\partial\xi} + \left( \frac{\partial \eta}{\partial x} \right) \frac{\partial u}{\partial\eta} \right] - \left[ \rho_e u_e \mu_e \frac{\partial\psi}{\partial\xi} + \left( \frac{\partial \eta}{\partial x} \right) \frac{\partial \psi}{\partial\eta} \right] \frac{\partial u}{\partial\eta} = -\sqrt{2\xi} \frac{\rho_e}{\rho} \mu_e \frac{\text{d}p_e}{\text{d}\xi} + \frac{\partial}{\partial\eta} \left( \frac{u_e \rho \mu}{\sqrt{2 \xi}} \frac{\partial u}{\partial\eta} \right)where the stream funciton, ψ, is defined by\frac{\partial\psi}{\partial y} = \rho u \qquad \frac{\partial\psi}{\partial x} = - \rho v
#Transform the dependent variables. Define a function of η and ξ, f(ξ,η), such that\frac{u}{u_e} = \frac{\partial f}{\partial y} \equiv f'where the prime denotes partial derivative with respect to η. From this equation, if follows that\frac{\partial u}{\partial \xi} = f' \frac{\text{d} u_e}{\text{d} \xi} + u_e \frac{\partial f'}{\partial \xi}\frac{\partial u}{\partial \eta} u_e f''
#Identify f with ψ. (missed remainder of this step, 7/31/2013)
#Obtain the final transformed equation. After substituting Eqns. 6.45-6.47 and 6.50 into Eqn. 6.43, and introducing Euler's equation (dpe = -ρeuedue) at the boundary layer edge. The x-momentum becomes(reproduce Eqn. 6.55)whereC = \frac{\rho u}{\rho_e u_e}The y-momentum equation in (ξ,η) space is\frac{\partial P}{\partial \eta} = 0The energy equation can be transformed as:(insert Eqn. 6.58 here)where g is a non-dimensional static enthalpyg = g(\xi,\eta) = \frac{h}{h_e}where he is the enthalpy at the boundary layer edge.Pr = \frac{\mu c_p}{k}
The transformed momentum equations and energy equation for the boundary layer are still partial differential equations. The transformed boundary layer equations must be solved subject to the following boundary conditions.
*At a wall for a fixed wall temperature,\eta = 0 \qquad f = f' = 0 \qquad g = g_wor for an adiabatic wallg' = 0
*At the boundary layer edge\eta \rightarrow \infty \qquad f' = 1 \qquad g = 1
Solution of Eqns. 6.55, 6.56, and 6.58, along with the appropriate boundary conditions yields,
:u = u_e f'(\xi,\eta)
and
:h = h_e g(\xi,\eta)
Pressure is known, since pe = pe(ξ) is known, and it is assumed that pressure is constant in the direction normal to the surface inside the boundary layer.
With h and p known inside the boundary layer, we can determine other state properties from equilibrium thermodynamics.
In Fig. 6.6, pg. 285, velocity and enthalpy profiles are shown for three different ξ. Even though these profiles are calculated in the transformed space, they are different for each ξ. This behavior is indicative of boundary layers that are non-self-similar. We will discuss these later.
The local skin friction coefficient, cf, is related to f" at the wall, and the local heat transfer rate at the surface is related to g'(ξ).
The local skin friction coefficient is
:c_f = \frac{\tau_w}{\frac{1}{2} \rho_e u_e^2}
where τw is the local shear stress at the wall,
:\tau_w = \left[ \mu \left( \frac{\partial u}{\partial y} \right) \right] _w
Combining the previous two equations and utilizing the transformation gives
:c_f = \frac{2\mu_w\rho_w}{\rho_e\sqrt{2\xi}} f(\xi,0)
The local heat transfer coefficient can be expressed in terms of the Nusselt number and the Stanton number.
:Nu = \frac{q_w x}{k_f (T_{aw} - T_w)} \qquad C_H = \frac{q_w}{\rho_e u_e (h_{aw} - h_w)}
For a calorically perfect gas
:Nu = C_H Re Pr
Eqns. 6.61 and 6.65 express the skin friction coefficient and the heat transfer coefficient at the wall in terms of f" and g'. The solution of Eqns. 6.55 and 6.58 for a general, non-similar boundary layer, requires the solution of coupled, non-linear partial differential equations as a two-point boundary-value problem. Currently we will consider cases where the boundary layer is self-similar.
===Flat-Plate Case===
For inviscid flow over a flat plat at zero angle of attack with constant properties,
(missed the last 15 minutes of class, 7/31/2013)
The shooting method is usually used to solve Eqns. 6.68 and 6.69, pg. 288. Eqn. 6.68 is third order, and Eqn. 6.69 is second order, so five boundary conditions must be specified. So far we only have three boundary conditions.
#f(0) = 0
#f(0)' = 0
#g(0) = gw
We need conditions for f"(0) and g'(0). The shooting method is executed as follows.
#Assume values for f"(0) and g'(0)
#Numerically integrate Eqns. 6.68 and 6.69. Integrate to large enough η such that f'(η) and g(η) are relatively constant in η.
#If f'(η) ≠ 1 and g(η) ≠ 1 for large η, assume new values for f"(0) and g'(0), and repeat steps 1 through 3.
For a flat plate, from Eqn. 6.63
:\xi = \rho_e u_e \mu_e x
For this case:
:C_f = \sqrt{2}\frac{\rho_w\mu_w}{\rho_e\mu_e} \frac{f(0)}{\sqrt{Re_x}}
:C_f \propto \frac{1}{\sqrt{Re_x}}
Note that for incompressible flow over at flat plate,
:C_f\text{(imcompressible)} = \frac{0.664}{\sqrt{Re_x}}
Regarding heat transfer to the flat plat
:C_H = \frac{1}{\sqrt{2}} \frac{1}{\mu_e} \frac{k_e}{c_{pw}} \frac{\rho_w}{\rho_e} \frac{h_e}{(h_{aw} - h_w)} \frac{g'(0)}{\sqrt{Re_x}}
:C_H \propto \frac{1}{\sqrt{Re_x}}
The incompressible value is given by
:C_H\text{(imcompressible)} = \frac{0.332}{\sqrt{Re_x}} Pr^{-\frac{2}{3}}
For the compressible case
:\frac{C_H}{C_f} = \frac{G}{F} = f\left( M_e,Pr,\gamma,\frac{T_w}{T_e} \right)
This is the Reynolds analogy. For the incompressible case,
:\frac{C_H}{C_f}\text{(imcompressible)} = \frac{1}{2}Pr^{-\frac{2}{3}}
A note on the viscosity and thermal conductivity coefficient. For a non-reacting gas, μ = μ(T). An engineering approximation is to assume an exponential temperature variation.
:\frac{\mu}{\mu_{ref}} = \left( \frac{T}{T_{ref}} \right) ^{\frac{3}{2}} \frac{T_{ref} + S}{T + S}
This is Sutherland's law where, for air
: \mu_{ref} = 1.789 \times 10^{-5} \frac{kg}{m\times s}
:T_{ref} = 288K
:S = 110K
and μ and T must have corresponding units. This is accurate for air over several thousand kelvins.
====Representative Results====
Velocity profiles for an adiabatic flat plate calculated by Van Driest are shown in Fig. 6.7, pg. 293. This figure shows that the boundary layer thickness becomes large at large Mach numbers.
Fig. 6.8, pg. 294, shows the drastic increase in temperature at large Mach numbers.
Fig. 6.9, pg. 295, and Fig. 6.10, pg. 296, are calculated for a cold plate. The effect of the cold wall is to thin the boundary layer. The cold wall causes a peak in temperature to occur within the boundary layer.
The coefficient of friction decreases with increasing Mach number. The effect of cooling the wall is to increase the Cf.
For an insulated wall, as Me → 0,
:C_f \sqrt{Re_x} \rightarrow 0.664
the incompressible value. CH follows the same trends.
====Summary====
#δ increases rapidly with Me
#T increases rapidly inside the boundary layer as Me increases, due to viscous dissipation
#Cooling the wall decreases δ
#Cf and CH decrease as Me increases
#Cf and CH increase as the wall is cooled
====Recovery Factor, r====
From Eqn. 6.63, pg. 286, we have
:q_w = \rho_e u_e C_H (h_{aw} - h_w)
We need to know what the adiabatic wall enthalpy, haw, is. In most engineering applications, the adiabatic wall enthalpy is expressed as,
:h_{aw} = h_e + r \frac{u_e^2}{2}
where r is the recovery factor.
From the definition of total enthalpy,
:h_0 = h + \frac{u^2}{2} = h_e + \frac{u_e^2}{2}
we can write
:h_{aw} = h_e + r(h_0 - h_e)
or,
: r = \frac{h_{aw} - h_e}{h_0 - h_e}
For a calorically perfect gas,
: r = \frac{T_{aw} - T_e}{T_0 - T_e}
For an incompressible flow, it can be shown that,
:r = \sqrt{Pr}
Note that r decreases as M∞ increases, see Fig. 6.13, pg. 299. For practical purposes, Eqn. 6.90 is adequate for hypersonic, laminar flow over a flat plate.
To obtain the wall heat transfer using Eqn. 6.85, pg. 298, CH is also needed, and it can be obtained from Eqn. 6.83, pg. 292.
The incompressible result which Anderson notes is a "reasonable approximation" for hypersonic laminar flow, is given by
:\frac{C_H}{C_f}\text{(imcompressible)} = \frac{1}{2}Pr^{-\frac{2}{3}}
===Stagnation-Point Case===
Consider the stagnation region on a blunt body, see Fig. 6.15, pg. 307.
*x is the distance measured along the surface
*R is the local radius of curvature
*ue is the velocity at the edge of the boundary layer in the x-direction
For the stagnation point, Eqn. 6.55 and 6.58 (momentum and energy equations) become
Momentum
:(Cf)' +ff = \frac{2\xi}{u_e} \left[ (f')^2 - \frac{\rho_e}{\rho} \right] \frac{\text{d}u_e}{\text{d} \xi}
and
Energy
:\left(\frac{C}{Pr}g'\right) ' + fg' = 2\xi \left[ \frac{f'g}{h_e} \frac{\partial h_e}{\partial \xi} + \frac{\rho_e u_e}{\rho h_e} f' \frac{\text{d}u_e}{\text{d} \xi} \right] - C\frac{u_e^2}{h_e} (f)^2 
In the stagnation region, ue is very small, so he (= h0) is very large, so
:\frac{u_e^2}{h_e} \approx 0
The flow in the stagnation region has a low velocity, so we can assume incompressible flow conditions in the flow near the stagnation point outside of the boundary layer. Therefore,
: u_e = \left( \frac{\text{d} u_e}{\text{d} x} \right) _s x
Following the transformation process used for the flat plate and assuming:
#a calorically perfect gas, and
#p = pe,
it can be shown that Eqns. 6.92 and 6.93, pg. 307, become:
Momentum
:(Cf)' + ff' = (f')^2 -g
Energy
:\left(\frac{C}{Pr}g'\right) ' + fg' = 0
These equations, 6.104 and 6.105, hold for a compressible, stagnation point boundary layer. They do not depend on ξ, and hence, this case is self-similar.
For a cylinder, Eqns. 6.104 and 6.105 lead to
:q_w = 0.57 Pr^{-0.6} (\rho_e \mu_e)^{0.5} \sqrt{\frac{\text{d}u_e}{\text{d}x}} (h_{aw} - h_e)
For an axisymmetric body it can be shown that
:q_w \propto \frac{1}{\sqrt{R}}
Hence, stagnation point heating varies inversely with the square root of the nose radius.
==Nonsimilar Hypersonic Boundary Layers==
We can use 3 methods for solving general, nonsimilar boundary layers.
===Local Similarity Method===
This method is not a precisely exact solution for general nonsimilar boundary layers. It is outlined as:
#Consider a boundary layer with properties at the outer edge and all that have an arbitrary variation with x.
#If the boundary layer is split into segments of thickness Δx that are small enough, the values of Tw, ue, he, pe can be assumed to not vary in each segment. Apply the transformed boundary layer equations (6.55 and 6.58) to each segment.
#Assume that all partial derivatives with respect to ξ are small enough to be neglected (approximation).
#Under these assumptions, Eqn. 6.55 becomes:(Cf)' + ff = \frac{2\xi}{u_e} \left[ (f')^2 - g \right] \frac{\text{d}u_e}{\text{d} \xi}and Eqn. 6.58 becomes:\left(\frac{C}{Pr}g'\right) ' + fg' = 2\xi \left[ \frac{f'g}{h_e} \frac{\partial h_e}{\partial \xi} + \frac{\rho_e u_e}{\rho h_e} f' \frac{\text{d}u_e}{\text{d} \xi} \right] - C\frac{u_e^2}{h_e} (f)^2 Eqns. 6.123 and 6.124 are approximate forms. They are ordinary differential equations that can be solved by the shooting method. (But they're not ODEs, see Eqn. 6.124)
===Difference-Differential Method===
This is an inherently exact solution of the general boundary layer equations. Equations 6.55 and 6.58 can be solved for an arbitrary boundary layer using a grid, as shown in Fig. 6.20, pg. 320.
In this method, ξ derivatives are replaced by finite differences quotients. Only derivatives in η appear in Eqns. 6.55 and 6.58.
Eqn. 6.55 becomes:
:(Cf)' + ff = \frac{2\xi}{u_e} \left[ (f')^2 - \frac{\rho_e}{\rho} \right] \frac{\text{d}u_e}{\text{d} \xi} + \frac{\xi}{\Delta\xi} \left[ 3(f')^2 + Af' - 3ff + Bf\right]
where A and B are known numbers from the previous boundary layer solutions at i-1 and i-2. Eqn. 6.58 is
:\left(\frac{C}{Pr}g'\right) ' + fg' = \frac{\xi}{\Delta\xi} (3f'g + Ef' - 3fg' + Fg') + 2\xi \frac{\rho_e u_e}{\rho h_e} f' \frac{\text{d}u_e}{\text{d} \xi} - C\frac{u_e^2}{h_e} (f)^2 + \frac{f'g'}{h_e}\frac{\text{d}h_e}{\text{d}\xi}
where E and F are known numbers similar to A and B.
These are ODEs that can be integrated across the boundary layer using the iterative shooting technique. No simplifications were made in going from Eqns. 6.55 and 6.58 to Eqns. 6.127 and 6.128, hence the difference-differential method is an exact method for the solution of general, nonsimilar boundary layers.
===Finite Difference Method===
In this method, both ξ and η derivatives are replaced with finite differences. Anderson notes that this is the state-of-the-art in hypersonic boundary layer solutions. The finite difference solution of a general, nonsimilar boundary layer is outlined as:
#Solutions must be started from a given solution at the leading edge, or at a stagnation point. This solution can be obtained from self-similar solutions.
#The next station downstream can be calculated using the approximate discretized forms of Eqns. 6.55 and 6.58, then the next can be calculated, and so on until the boundary layer solution is obtained across the whole boundary layer.
#Then the skin friction and heat transfer at the wall can be calculated.\tau = \left[ \mu\left( \frac{\partial u}{\partial y} \right) \right] _wandq = \left[ k \left( \frac{\partial T}{\partial y} \right) \right] _wwhere one-sided differences are used.
An example of the finite difference boundary layer solution is shown in Fig. 6.22 and 6.23, pg. 325-326, for an axisymmetric hyperboloid.
==Hypersonic Transition==
"Nature left to its own devices always moves to a state of maximum disorder." Viscous flow is a perfect example. They start as orderly laminar flow, but at some location downstream, they transition to disorderly, turbulent flow.
Turbulence, and transition to turbulence, is an unsolved problem. Therefore, engineers treat the problem as approximate and depend on empirical data, especially at hypersonic speeds where transition behavior strays from that realized at lower speeds.
Consider flow over a flat surface, shown in Fig. 6.34, pg. 328. The flow starts at the leading edge as laminar. Laminar flow is highly stable and does not amplify disturbances. At some point downstream, the flow becomes unstable and flow disturbances are amplified. This is the start of transition. As amplification of disturbances continues, transition to turbulence takes place and continues until the flow is fully turbulent. This is the end of the transition region.
Our knowledge about transition is very imprecise. We cannot predict the transition region, so we usually assume it takes place at a single point, xT.
The transition Reynolds number is defined as:
:Re_T = \frac{\rho_e u_e x_T}{\mu_e}
ReT is critical for accurate prediction of skin friction and aerodynamic heating. There is no accurate theory for the prediction of xT (and therefore ReT), so we must rely on experimental data.
To give a feel for the complexity of the problem, Anderson presents the results from Stetson, who analyzed the empirical database. Stetson presented ReT as:
:Re_T = f\left( M_e,\theta_c,T_w,\dot m,\alpha, k_R, E,\frac{\partial p}{\partial x},R_N,Re_\infty /ft, \frac{x}{R_N}, V, C, \frac{\partial w}{\partial z}, T_0,d^*,\tau,Z \right)
These 17 parameters, upon which ReT depends, are identified in the text, pg. 329. We will cover some of the more significant parameters.
===Mach Number===
Stability theory shows that the stability of the laminar boundary later is generally enhanced by larger Mach numbers at the boundary layer. See Fig. 6.25, pg. 329.
For incompressible flow on a flat plate, transition usually occurs at ReT ~5×105. For hypersonic flow, transition usually occurs at higher Re. It can be ~108. This is beneficial because heating and friction are considerably smaller for laminar flow. This delayed onset of transition leads to larger areas of laminar flow on hypersonic bodies.
===Environment===
Transition is sensitive to environmental disturbances such as
*Free stream turbulence
*Acoustic disturbances
*Turbulent boundary layer on wind tunnel walls (Anderson says that "wind tunnel measurements are always compromised.")
Anderson says that "meaningful transition data must be obtained from free-flight experiments." This is a pretty strong statement that seems to brush aside work done on quiet wind tunnels.
===Unit Reynolds Number===
Unit Reynolds number is the Reynolds number based on a unit length. There is no physical explanation for the unit Reynolds number to influence transition, but experimental evidence shows that there is a correlation.
===Angle of Attack===
3-D effects have a strong influence on transition. Non-zero angles of attack result in asymmetric 3-D flow.
An example of measured transition on a cone (θc = 8°) is shown in Fig. 6.26, pg. 331. As α is increased, transition moves rearward on the windward side and forward on the leeward side.
The effect of azimuthal position on an 8° cone (at α = 2° and M∞ = 6) transition region is shown in Fig. 6.27, pg. 332. The transition region moves upstream as we move from the windward to leeward side of the cone.
===Nose Bluntness===
We know that the blunt body will produce a bow shock wave that creates an entropy layer downstream of the nose. An example of an inviscid flow calculation for a blunted 8° cone at α = 0° is shown in Fig. 6.28, pg. 333. Note that the nose radius, RN, is only 0.04 in and the cone length is 14 in. Even with this small nose radius, M and Re/ft vary strongly and do not recover the sharp cone values until the end of the cone. Pressure recovers much more quickly.
The addition of the blunted nose, even for small nose radius on the cone, delays transition. See Fig. 6.29, pg. 334. Note that for large RN, transition can occur prematurely on the nose. This nose-tip transition is often referred to as "the blunt-nosed paradox." This occurs in spite of strong, favorable pressure gradients. This contradicts the case of the laminar boundary layer where favorable pressure gradients stabilize the boundary layer.
===Wall Temperature===
For moderate wall cooling, the hypersonic boundary layer is stabilized, as observed in the low-speed case. However, for the high wall cooling case, there is evidence of reversal of the trend, where ReT decreases.
===Prediction of Transition===
Even though it is highly approximate, we need to be able to calculate ReT. One method for hypersonic transition is based on the boundary layer momentum thickness, θ, where θ is defined as:
:\theta = \int_0^\delta \frac{\rho u}{\rho_e u_e} \left( 1- \frac{u}{u_e} \right) \text{d}y
Then
:Re_{\theta_T} = \frac{\rho_e u_e \theta_T}{\mu_e}
For some cases, the following relation, based on hypersonic empirical evidence, has been used.
:Re_{\theta_T} = 100 M_e
Another prediction correlation, based on cone data, is:
:\log_{10}(Re_T) = 6.421 \exp [1.209 \times 10^{-4} M_e^{2.641} ]
This correlation has been used in the study of hypersonic wave riders.
The estimates are far from appropriate for applications to all hypersonic transitions. However, we must have a value for design purposes, and something (Eqns. 6.139 and 6.140) are better than nothing.
==Hypersonic Turbulent Boundary Layer==
There is no precise way to analyze the turbulent boundary layer, so empirical data are required, and substantial uncertainty can result. Large-scale turbulent motion leads to energy being transmitted more readily than in laminar flow. This leads to:
#Fuller velocity profiles through a turbulent boundary layer
#Larger velocity gradients at the wall
#Skin friction and heat transfer are larger
In the previous list, 2 is a result of 1 and 3 is a result of 2.
The same results occur for low speed flow. It is important to have a model for turbulent flow. This is a state-of-the-art subject.
Anderson discusses the Baldwin-Lomax turbulence model. This model:
#is a typical, engineering-oriented model
#has been used in many supersonic and hypersonic flows
#is an eddy viscosity model, where the effects of turbulence in governing equations are included by simply adding a term to the transport coefficients.
Consider the x-momentum boundary layer equation for turbulent flow.
:\rho u \frac{\partial u}{\partial x} + \rho v \frac{\partial u}{\partial y} = - \frac{\partial P}{\partial x} + \frac{\partial}{\partial y}\left[ (\mu + \mu_t) \frac{\partial u}{\partial y} \right]
Here, the viscosity, μ, has been augmented by a turbulence component, μT.
The Baldwin-Lomax model utilizes just algebraic relations only involving the flow properties. This model is highly empirical, and it yields reasonable results across a wide range of Mach numbers, from subsonic to supersonic. This model splits the boundary layer into two layers with different μT for each.
:\mu_T = \begin{cases} (\mu_T)_{inner}, & y\le y_{crossover} \\ (\mu_T)_{outer}, & y > y_{crossover} \end{cases}
where y is the normal distance from the wall, and ycrossover represents the location of the crossover point.
For the inner region,
:(\mu_T)_{inner} = \rho l^2 |\omega |
where,
:l = ky\left[1 - \exp \left(\frac{-y^+}{A^+} \right) \right]
:y^+ = \frac{\sqrt{\rho_w \tau_w} y}{\mu_w}
and k and A+ are dimensionless constants, and ω is the local vorticity.
For the outer region,
:(\mu_T)_{outer} = \rho KC_{cp}F_{wake}F_{Kleb}
where K and Ccp are two additional constants, and Fwake and FKleb are related to the function
:F(y) = y |\omega | \left[1 - \exp \left(\frac{-y^+}{A^+} \right) \right]
F(y) will have a maximum, Fmax, which occurs at ymax. Fwake is taken to be
:F_{wake} = \min \left( F_{max} y_{max}, \frac{C_{wk} y_{max} U_{dif}^2}{F_{max}} \right)
where Cwk is a constant, and
:U_{dif} = \sqrt{u^2 + v^2}
FKleb is the Klebanoff intermittency factor, given by
:F_{Kleb}(y) = \left[ 1+5.5\left( C_{Kleb}\frac{y}{y_{max}} right) ^6 \right] 
The dimensionless constants are:
*A+ = 26.0
*Ccp = 1.6
*CKleb = 0.3
*Cwk = 0.25
*k = 0.4
*K = 0.0168
The Baldwin-Lomax model is based on local vorticity, ω. It is independent of the flow field properties themselves.
The molecular values of viscosity and thermal conductivity are related through the Prandtl number.
: k = \frac{\mu c_p}{Pr}
Instead of developing a detailed turbulence model for thermal conductivity, kT, a turbulent Prandtl number is defined.
: k_T = \frac{\mu_T c_p}{Pr_T}
where the usual assumption is that PrT = 1.
===Typical Results===
Consider hypersonic turbulent flow over a flat plate. Results for the variation of Cf with Mach number as given in Fig. 6.30, pg. 339. The solid curve is the Van Driest solution, the standard solution for comparison. It is within 10% of experimental data. Cf decreases with free stream Mach number. This is stronger for turbulent flows than for laminar flows.
The heat transfer, CH, results are shown in Fig. 6.31, pg. 340. Lines for both turbulent and laminar flows are shown. For a given Re, the Mach number effect is stronger for the turbulent flow. For a given Mach number and Re, the turbulent values for CH are considerably higher.
==Reference Temperature Method==
==Hypersonic Aerodynamic Heating: Some Comments and Approximate Results Applied to Hypersonic Vehicles==
==Entropy-Layer Effects on Aerodynamic Heating==
==Summary and Comments==