Hypersonic and High-Temperature Gas Dynamics Chapter 5 Notes

Chapter 5 is titled "Hypersonic Inviscid Flowfields: Exact Methods."
==General Thoughts==
Chapter 4 covered "classical" hypersonic aerodynamics, developed in the 1950s and 1960s (before widespread use of computers). The approximate methods, e.g. small perturbations, are taken as limiting cases (small θ, β, and M∞ → ∞).
In this chapter, solution of the exact Euler equations for inviscid flow without approximations will be presented. This chapter deals with the application of CFD to inviscid hypersonic flows. Note the importance of CFD in hypersonic aerodynamics is due to limited access to hypersonic wind tunnels and other ground test facilities due to:
*Number of tunnels
*Flight Mach number range
*Reynolds number
*Attainable temperature range
Computational techniques include:
#Method of Characteristics (MoC)
#Finite difference methods
#Finite volume methods
#Finite element methods
==Method of Characteristics==
The method of characteristics was first applied to the design of the supersonic nozzle in 1929. It has become a classical technique for the solution of supersonic and hypersonic flows. The MoC can be applied when the system of governing equations is hyperbolic.
The governing equations for steady, inviscid, supersonic (and hypersonic) flow are hyperbolic. The flow field downstream of the limiting characteristic (to be defined later) can be determined by marching downstream along characteristic lines (2-D flow) or characteristic surfaces (3-D flow).
The Euler equations reduct to simpler differential equations in one less space dimension, called the "compatibility equations." The compatibility equations are ODEs in 2-D flow and PDEs in 3-D flow.
The simplest application of the MoC involves steady, 2-D, irrotational flow. For this case there are two characteristics at each point in the flow, which are left-running and right-running waves. The compatibility equations along these characteristics are algebraic relations. For 2-D, steady, irrotational, axisymmetric flow, the characteristics are ODEs.
If the flow is steady, 2-D or axisymmetric, rotational flow, there are 3 characteristics through each point (left- and right-running Mach waves and the streamline).
The compatibility equations are ODEs that hold along these characteristics.
For 3-D flows, the characteristics are Mach surfaces, and stream surfaces. The compatibility equations are PDEs in two spatial dimensions. For application of MoC to hypersonic flows, rotationality is a complication. Recall that the flow behind curved shocks is rotational.
Crocco's theorem for steady inviscid flow is:
:T\nabla s = \nabla h_0 - \vec V \times (\nabla \times \vec V)
:\nabla h_0 \equiv \text{gradient in total enthalpy across the shock} = 0
:\nabla \times \vec V \equiv \text{vorticity}
Vorticity is non-zero for curved shock, since the strength, and therefore entropy change associated with the shock, change with the position along the shock. Hypersonic flows can be highly rotational; therefore, rotational MoC is required.
Consider the application of the rotational MoC to a 2-D or axisymmetric external flow. The usual method is described below.
#Start from an initial data line that must be obtained from another independent calculation.
##If the body has a pointed nose with an attached shock wave, then the flow close to the nose is supersonic and can be approximated as a wedge (2-D) or conic (axisymmetric) flow.
##If the body has a blunt nose with detached shock wave, an appropriate blunt-nosed solution must be found. Note that the data line must be taken along or downstream of the limiting characteristic (the characteristic that touches the sonic line at only one point, and hence does not cross into the subsonic region). See Fig. 5.7b, pg. 188. Also note that a characteristic line cannot be used as the initial data line.
#March downstream from the initial data line along the characteristic lines. Referring to Fig. 5.8, pg. 189, points 1 and 2 are on the initial data line. All properties are known along this line. Since M1 and M2 are known, the Mach angles μ1 and μ2 are known. The left-running (C+) and right-running (C-) Mach waves are constructed at both points. C+ and C- are characteristic lines and intersect at point 3.
#Assume the streamline at point 3 is at angle \theta_3 = \frac{1}{2}(\theta_1 + \theta_2). Trace the streamline through point 3 back until it intersects the initial data line (at point 4). For rotational flow, the streamline is also a characteristic line.
#Calculate the flow properties at point 3 by solving the compatibility equations along the characteristic lines. After a lot of manipulation of Euler's equations, the compatibility equations are:
*Along Mach lines:\frac{\text{d}p}{\rho V^2 \tan \mu} \pm \text{d}\theta + \frac{j\sin\theta \sin\mu}{\sin(\theta \pm \mu)} \frac{\text{d} y}{y} = 0
*Along Streamlines:\text{d}s = 0
Where, the plus and minus signs refer to the C+ and C- characteristics and j = 0 or 1 for two-dimensional and axisymmetric flows, respectively.
The streamline equation can be replaced by two relations along the streamline with constant entropy.
:\text{d}p = -pV\text{d}V (Euler's equation along a streamline in rotational flow)
and
:\text{d}p = a^2\text{d}\rho (definition of speed of sound)
because s is constant along the streamline.
Note that dp in Eqns 5.4 and 5.5 denotes the change in pressure along a streamline; dp in Eqn. 5.2 denotes a change in pressure along the Mach lines.
(Missed the first 50 minutes of class, 7/24/2013)
==Correlations for Hypersonic Shock-Wave Shapes==
==Time-Marching Finite Difference Method: Application to the Hypersonic Blunt Body Problem==
All preceding discussion in this course was state-of-the-art in hypersonic aerodynamics prior to 1966. In 1966 Moretti and Abbett published the first practical blunt-body solution obtained with a time-marching finite difference technique.
Why was the hypersonic blunt-body problem so hard to solve? The shock wave in front of the blunt body is detached and curved, ranging from a normal shock wave at the nose to a weak Mach wave at large distances from the body (see Fig. 5.19, pg. 201). Parts of the flow behind the shock are subsonic; hence, the blunt-body shock layer is mixed subsonic and supersonic flow.
*The steady, subsonic region is elliptic. As a result, the flow at a given point depends, simultaneously, on the flow properties at all other points in the subsonic region.
*The steady, supersonic region is hyperbolic. The nature of hyperbolic problems is that flow at any give point only depends on the properties at other points bounded by Mach lines/surfaces reaching upstream from that point.
*Theoretical/numerical techniques properly posed for one region are ill-posed for the other.
The approach utilized by Moretti and Abbett is a "time-marching finite difference" solution governed by unsteady Euler equations. Note that the unsteady Euler equations (Eqns. 4.1-4.5) are hyperbolic with respect to time, irregardless of whether the flow is subsonic or supersonic. We will follow Moretti and Abbett; however, we will use a simpler version that was developed by MacCormack, the explicit predictor-corrector finite difference method.
We will assume a 2-D flow. The time-dependent Euler equations for this flow are:
Continuity
:\frac{\partial \rho}{\partial t} = -\left[ \frac{\partial (\rho u)}{\partial x} + \frac{\partial ( \rho v)}{\partial y} \right]
X-Momentum
:\frac{\partial u}{\partial t} = -\left[ u\frac{\partial u}{\partial x} + v\frac{\partial u}{\partial y} + \frac{1}{\rho}\frac{\partial p}{\partial x} \right]
Y-Momentum
:\frac{\partial v}{\partial t} = -\left[ u\frac{\partial v}{\partial x} + v\frac{\partial v}{\partial y} + \frac{1}{\rho}\frac{\partial p}{\partial y} \right]
Energy
:\frac{\partial}{\partial t} \left( \frac{p}{\rho^\gamma} \right) = -\left[ u\frac{\partial}{\partial x} \left( \frac{p}{\rho^\gamma} \right) + v\frac{\partial}{\partial y} \left( \frac{p}{\rho^\gamma} \right) \right] 
To solve these equations for a blunt body in 2-D flow, the following steps are performed:
#We have a given body shape.
#Assume the shock wave shape and the shock wave stand-off distance. Generate a discrete grid between the shock and the body (see Fig. 5.20a, pg. 204).
#Assume flow field variables p, u, v, and ρ at each grid point (initial conditions).
#Calculate the flow field at next time step by using a finite difference solution of the governing equations.Because the finite difference approach is easier to apply to a rectangular grid, the curvilinear physical space can be transformed into a rectangular space using:\zeta = \frac{x-b}{\delta}where δ is the shock stand-off distance and b is the body position. In the transformed space, the shock is at ζ = 1 and the body is at ζ = 0. See Fig. 5.20, pg. 204, for more details.
#For convenience, transform the dependent variables:P = \ln p \qquad R = \ln \rho \qquad \psi = \ln p - \gamma \ln \rho = P - \gamma RAlso let,C \equiv (\zeta - 1) \frac{\text{d}b}{\text{d}y} - \zeta \cot \thetaW \equiv \frac{\text{d}s}{\text{d}t} \quad x-component of the shock wave velocityB = \frac{u - W\zeta + vC}{\delta}In terms of these variables, the governing equations become:Continuity\frac{\partial R}{\partial t} = -\left[ B\frac{\partial R}{\partial \zeta} + \frac{1}{\zeta} \frac{\partial u}{\partial \zeta} + \frac{C}{\zeta} \frac{\partial v}{\partial \zeta} + \frac{\partial v}{\partial y} + v \frac{\partial R}{\partial y} \right]X-Momentum\frac{\partial u}{\partial t} = -\left[ B\frac{\partial u}{\partial \zeta} + v\frac{\partial u}{\partial y} + \frac{p}{\rho \delta}\frac{\partial p}{\partial \zeta} \right]Y-Momentum\frac{\partial v}{\partial t} = -\left[ B\frac{\partial v}{\partial \zeta} + v\frac{\partial v}{\partial y} + \frac{pC}{\rho \delta}\frac{\partial P}{\partial \zeta} + \frac{p}{\rho}\frac{\partial P}{\partial y}\right]Energy\frac{\partial \psi}{\partial t} = -\left[ B\frac{\partial \psi}{\partial \zeta} + v\frac{\partial \psi}{\partial y} \right] Once the flow field variables are obtained at the grid points, in the computational space, they can be carried to the corresponding grid points in the physical space. Note that the grid in physical space is changing in time since the position and the shape of the shock wave are changing, as a steady-state solution is approached.
#Calculation of the flow field — illustrated for the x-component of velocity. The velocity from the previous time step is know, so,u_{i,j}^{t+\Delta t} = u_{i,j}^{t} + \left( \frac{\partial u}{\partial t} \right) _{ave} \Delta twhere, \left( \frac{\partial u}{\partial t} \right) _{ave} is the average time derivative of u between t and t + Δt (the predictor-corrector philosophy).
#Calculate \left( \frac{\partial u}{\partial t} \right)_{i,j}^t from x-momentum equation using forward differences in space.\frac{\partial u}{\partial t} = -\left[ B_{i,j}^t\frac{u_{i+1,j}^t - u_{i,j}^t}{\Delta \zeta} + v_{i,j}^t\frac{u_{i,j+1}^t - u_{i,j}^t}{\Delta y} + \left( \frac{p}{\rho \delta} \right) _{i,j}^t\frac{P_{i+1,j}^t - P_{i,j}^t}{\Delta \zeta} \right]
(Missed steps 8 to 12, 7/24/2013)
We need to have proper boundary conditions for the:
*Shock wave
*Body
*Downstream
*Centerline (axisymmetric case)
===Shock Wave Boundary===
The shock wave is treated as a discontinuity, across which local shock wave relations apply. Anderson's approach is to calculate the flow properties at a grid point at time t + Δt by applying the internal flow algorithm (steps 6-11), with one modification: at the shock grid points a rearward difference must be used in both the predictor and corrector steps. The particular pressure at t + Δt is obtained.
It is important to note that the shock wave is moving until a steady solution is reached. The value of W obtained at each time step is used to adjust the position of the shock wave.
===Body Boundary Condition===
ζ = 0 along the body. The flow must be tangent to the surface, i.e.
:\vec V \cdot \hat n = 0
Anderson's approach is to use one-sided differences (forward differences) in the predictor-corrector steps at the body. This allows steps 6-11 for the internal flow algorithm at the body.
In general \vec V will not be tangent to the body. To make Vn = 0, we will send out a local, finite, 1-D isentropic expansion or compression wave of sufficient strength to cancel Vn.
*When Vn is toward the surface, a compression wave is used to induce a mass motion in the direction of the wave. See right side of Fig. 5.22, pg. 211.
*When Vn is away from the surface, a finite expansion wave is emitted to induce a mass motion opposite to the direction of the wave, thus canceling the outgoing Vn. See left side of Fig. 5.22, pg. 211.
===Downstream Boundary Condition===
At the downstream boundary (the top of the grid in Fig. 5.20b, pg. 204), the flow properties are obtained from linear extrapolation from the values at the adjacent internal grid points. This is sufficient as long as the downstream boundary condition is supersonic all along the boundary.
===Centerline Boundary Condition===
For a 2-D or axisymmetric flow, at zero angle of attack, the centerline is a line of symmetry.
:\frac{\partial p}{\partial y} = \frac{\partial T}{\partial y} = \frac{\partial u}{\partial y} = 0
or,
:p_{j+1} = p_j \quad T_{j+1} = T_j \quad u_{j+1} = u_j 
Because v changes sign at the centerline,
:v_{j+1} = -v_j
===Stability===
Since the finite difference approach presented is explicit, it is subject to stability criterion. Δt cannot exceed a certain value to maintain a numerically-stable solution. This is the Courant-Friedrich-Lewy (CFL) criterion, and for our 2-D flow it is given by:
:\Delta t \le min(\Delta t_x, \Delta t_y)
where,
:\Delta t_x = \frac{\Delta x}{u+a} \quad \text{and} \quad \Delta t_x = \frac{\Delta y}{v+a}
Δtx and Δty are the time it takes an acoustic wave to travel between two adjacent grid points in the x- and y-directions, respectively.
===Typical Results===
Typical results for hypersonic blunt-body flows obtained by means of the time-marching scheme are given in Figs. 5.24-5.27, pg. 214-216, for flow over a parabolic cylinder at zero angle of attack.
*Fig. 5.24, pg. 214, shows the evolution of the shock wave with time. The change reduces as time progresses.
*Fig. 5.26, pg. 215, shows a comparison of the exact surface pressure distribution with solutions using the Modified Newtonian approach for the parabolic cylinder.
*Fig. 5.27, pg. 216, shows a comparison of the surface pressure distribution of the exact numerical results and the Modified Newtonian results for a paraboloid.
Recall that by definition, the limiting characteristic is the locus of points which has only one point of the sonic line in its "zone of action." In other words, the limiting characteristic only touches the sonic line at one point. See Fig. 5.29, pg. 217. Only when we move downstream of the limiting characteristic will the left running waves no longer intersect the sonic line.
What is the physical implication of this?
Although the flow downstream of the sonic line is supersonic, disturbances produced in this region will propagate to the sonic line and can affect the entire subsonic region. This is why the limiting characteristic must remain upstream of the data line in MoC.
===Initial Condition for Non-Zero Angle of Attack===
Returning to the entropy equation (Eqn. 4.5),
:\frac{\text{D}s}{\text{D}t} = 0
At a stagnation point, where velocity is zero, this becomes:
:\frac{\partial s}{\partial t} = 0
So, at a stagnation point in unsteady, inviscid flow, the entropy remains constant (independent of time). Therefore, for a time-marching scheme, the stagnation point conditions cannot arbitrarily be chosen. The proper steady-state value of entropy must be used, because it will not change. However, this is no problem for the symmetric case (zero angle of attack) because we know the steady state conditions are identical to the stagnation conditions behind a normal shock. For the asymmetric case, chances are that none of the grid points (on the surface) will correspond to the exact location of the stagnation point, and the problem is not encountered.
==Shock-Shock Interactions==
Shock wave interaction will occur when one shock impinges on another. Some of these interactions are straight-forward, i.e. those involving straight oblique shock waves which can be calculated using classical compressible flow techniques. Others are complex, involving curved shock waves which produce mixed subsonic-supersonic flow, i.e. shock wave interactions typical of hypersonic flow.
Shock-shock interactions must be considered in the design of hypersonic vehicles. If not taken into account, parts of the vehicle can be destroyed. See Fig. 5.37, pg. 226, for the depiction of the shock interaction at the cowl of the X-43. Type 4 shock interaction is shown in the inset of the figure. The inlet bow shock and cowl shock interact to form a complex flow that results in a supersonic jet directed toward the cowl.
There are six types of shock-shock interaction that are depicted in Fig. 5.38, pg. 227. In this figure, a blunt body is in hypersonic flow. The resulting bow shock is labeled BS, the incident shock is labeled IS, and the transition from subsonic to supersonic flow is labeled SP.
The interaction of IS and BS will result in a flow characterized by six different patterns that depend on the sector in which the interaction occurs. These are described below:
#Interaction of two straight oblique shocks of opposite families. This results in two transmitted shocks and a slip line trailing downstream from the intersection. The flow is locally supersonic everywhere and can be calculated using classical compressible flow analysis.
#The IS impinges just downstream of the SP behind the BS. This results in two intersecting shock waves connected by a nearly normal shock (Mach reflection). This interaction has mixed subsonic-supersonic flow that must be calculated by a time-marching numerical technique. This produces a subsonic region between supersonic regions, separated by two slip lines.
#Here the IS impinges on the BS just upstream of the SP. The flow behind the BS is subsonic. A slip line trails downstream from the intersection point and impinges on the body. The flow above the line is subsonic, and the flow below the line is supersonic. The supersonic flow is turned along the body by a series of compression and expansion waves.
#The IS impinges on the stronger part of the BS, far upstream of the sonic point, where the turing angle of the flow is too large for an attached shock. Behind the shock impingement point, a supersonic jet is formed, bounded on both sides by slip lines. This jet is slowed by a series of oblique shocks that terminate in a normal shock close to the body. Huge spikes in pressure and aerodynamic heating result. Heating rates up to 30x those without shock interactions can occur (results from both numerical simulations and experiment).
#The IS and BS are of the same family. The intersecting waves are connected by a normal shock. This normal shock is essentially a Mach reflection.
#The IS and BS are of the same family and are located far downstream of the sonic point on the body. Two supersonic flows result downstream separated by a slip line
==Space-Marching Finite Difference Method: Additional Solutions of the Euler Equations==
==Comments on the State of the Art==
==Summary and Comments==