5.0 Aerothermodynamics
Giles Goetz
5.1 Nomenclature
K = Proportionality constant used in the modified Newtonian theory
K = 2, from Reference 2
S = Arbitrary Reference Area set to be the projected area of the cylinder with radius R.
S = R2
Arbitrary reference length, set to be the length of the cylinder, L along the x-axis.
= Angle of attack, the angle between the vehicle and the velocity vector
= Fin angle, the angle between the horizontal of the fin and the horizontal of the vehicle body,
L = Length of the cylinder
R = Radius of the cylinder and the hemisphere
Le = Length of the leading edge of the fin.
Re = Radius of the leading edge of the fins.
= Sweep angle of the fin
Lf = Length of the fin, dependent on the length of the leading edge and the sweep angle
Wf = Width of the fin, dependent on the length of the leading edge and the sweep angle
Xcg = Distance along the x-axis from the reference point to the center of gravity with positive direction being forward.
Zcg = Distance along the z-axis from the reference point to the center of gravity.
All the coefficient equations are non-dimensionalized by the reference area and reference length in the case of the moment coefficients.
5.2 Vehicle Parameters and Design
The vehicle used for the case studies is blunt with a low L/D ratio. A diagram of the vehicle can be seen below in Figure 5.1. The coordinate system for the vehicle is labeled below in Figure 5.1 and has the variables x, y and z for the three dimensions. The positive directions are labeled with the bold arrows.
Figure 5.1: Diagram of Vehicle with Coordinate System
The different vehicle parameters can be seen in Figures 5.2, 5.3 and 5.4. The nose is a hemisphere with radius R, and is attached to a cylindrical body of length L with same radius R. A set of control fins are attached perpendicular to the body. The control fins are of length Lf and width Wf. The control fins have a leading of length Le, radius Re and a sweep angle . The different vehicle parameters are described in detail later.
5.3 Methods
5.3.1 Aerodynamics
The aerodynamics of the vehicle was modeled using a combination of methods. The different methods were hypersonic Newtonian theory, hypersonic skin friction, supersonic skin friction, supersonic linearized flow, wave drag and viscous interaction effects. The results from the various methods were used to form a model of the vehicle to get Cl, Cd, and Cm for a given velocity, angle of attack, and a set of vehicle parameters. The different parameters used can be found in Table 5.1 in section 5.3.
Hypersonic Newtonian Theory
Newtonian Theory uses the pressure distribution of air flowing over a body to calculate forces and moments at hypersonic speeds. The equations used for the Newtonian theory came from Reference 3. The method used was to take simple shapes and combine them to get an overall model of the vehicle. Clark and Trimmer’s paper contained base shapes along with their Cl, Cd and Cm calculations. By using the reference points and equations for each shape, it was possible to combine the shapes to create an aerodynamic model of the vehicle.
Hemisphere
Figure 5.2: Diagram of a Hemisphere with Reference Point
The following equations are for the Cn,Hemi and Ca,Hemi for a hemisphere of radius R with a reference point at the center of curvature. The moment coefficients for a hemisphere are zero about the center of curvature. The equations can be found in Reference 3, pp 28-29.
(Ref 3, Equation 120)
(Ref 3, Equation 121)
Cylinder
Figure 5.3: Diagram of Cylinder with Reference Point
The following equations are for the Cn,Cylinder, Ca,Cylinder and Cm,Cylinder for a cylinder of radius R with a reference point at the end. The equations can be found in Reference 3, pp 34.
(Ref 3, Equation 155)
(Ref 3, Equation 156)
(Ref 3, Equation 157)
Fins
Figure 5.4: Diagram of Fins and Reference Point
The Newtonian calculations treat the leading edges of both fins as a single unit for analysis. The equations for the leading edges are from Reference 3, pp 12-13.
(Ref 3, Equation 38)
(Ref 3, Equation 42)
(Ref 3, Equation 43)
It is also necessary to include the effect of the flat plate part of the fins. The equations for coefficients of the flat plate come from Reference 1, pp 51. At the center of pressure, the Cm,Plate is equal to zero, and flat plates do not have any axial force. Since the center of pressure is not the reference point of the fins, the Cm,Plate of the flat plate part of the fins needs to be transferred to the reference point of the fins. The distances from the center of pressure to reference point are used to calculate Cm,Plate at the reference point with Equation 161, from Reference 3, pp 35. The CN,Plate value for the flat plate part is also multiplied by two to take into account two fins. The two in front of the equation for CN,Plate in Anderson’s book, is the K factor, and not the two representing the two fins.
(Ref 1, Equation 3.9)
(Ref 2, pp 51)
(Ref 3, Equation 161)
Final Composite Configuration
In order to combine everything, need to first make sure everything is in the right frame. Since the fins can move relative to the body, a coordinate transformation must be made. The fins are related to the coordinate system of the vehicle body with the angle . is the pitch angle between the fins and the x-axis of the body. = 0 when the fins are parallel to the x-axis of the vehicle body and the z-axis points straight up through the fins, and = 90 when the fins are parallel to the z-axis and the x-axis points through the bottom of the fins. Using the transformation equations from Reference 1, pp 17, Equations 1.1 and 1.2, the coefficients for the parts of the fins were added and applied to a coordinate transfer to the body frame, shown in the following equations. Also note that since the two plate’s moment coefficients were about their centers of pressure, the coefficients of moment had to be moved to the reference point of the body by use of Reference 3, Equation 161, pp 35.
(Fin Equation 1)
(Fin Equation 2)
(Fin Equation 3)
Now that all the parts are in the same frame, it is possible to just add the normal and axial coefficients. The moment coefficients for the hemisphere and fins need to be translated to the reference axis of the whole vehicle using Equation 161, Reference 3, pp 35. These final composite equations can be seen below.
Finally, need to shift the center of the moment coefficient to the center of gravity of the vehicle. Again, this is done using Reference 3, Equation 161.
Once the coefficients are known, Cl and Cd are calculated using Equations 1.1 and 1.2, Reference 1, pp 17.
Hypersonic Skin Friction
Since Newtonian theory only takes into account the shape of the vehicle, other techniques are needed to further analyze the effects of the flow around the vehicle. The first such method is hypersonic skin friction. Using Reference 4, the process was coded as follows.
First the Mach number is determined from the velocity. This indicates if the flow is hypersonic or supersonic. The subsonic case will not be analyzed for the Mars entry vehicle. Next, the local edge Mach number is calculated by using the crude relation of Me = Mcos(). Now the location of transition on the surface of the vehicle is found by calculating the Reynolds number at that location by using Equation 72, Reference 4, pp 35. Finally the location of the transition is found by using Equation 71, Reference 4. In Equation 71, ’ is the reference viscosity, found using the Sutherland law, ’ is the reference density and can be found from the reference temperature, which is calculated from the wall temperature and free stream Mach number and temperature. This is an important step because the value for the hypersonic skin friction is dependent on whether the flow is turbulent or laminar.
Now the Cd,hyper can be found using Equation 86 on page 37 of Reference 4. In Equation 86, Re’l is Reynolds number at the end of the vehicle using reference density and viscosity, Re’x,t is the Reynolds number at transition using reference density and viscosity, Rex,t is the Reynolds number at transition using free stream density and viscosity, xt is the point on the body where transition occurs, T’ is the reference temperature, T is the free stream temperature and l is the length of the vehicle which is computed from the sum of the arc length of the hemisphere, and the length of the cylinder.
The process is then duplicated for the fins, substituting in the different lengths, and instead of using , the code uses the angle + to represent the angle of attack of the fins. Then the two hypersonic skin friction values are added together to generate a total hypersonic skin friction value.
Supersonic Skin Friction
Since the vehicle will not always remain at hypersonic speeds, calculations for skin friction at supersonic speeds must be made. Again, the Reynolds number and position of transition are found using Equations 71 and 72 from Reference 4. Now the incompressible skin friction is found using Equation 88 from Reference 4, on page 38, and then factoring in compressibility by using Equation 89. Finally Cd,super is calculated by using Equation 91 on page 39 of Reference 4. Instead of multiple pieces, the CF for just the vehicle is used, with the Sref being the projected area of the cylinder, and the Sfuselage being the surface area of the vehicle. The supersonic skin friction is also calculated for the fins using the surface area of the fins, and the same reference area for the body. The two values are then added to get a total supersonic skin friction value.
Supersonic Linearized Theory
Since the hypersonic equations do represent the vehicle well at lower supersonic speeds, supersonic linearized theory was used to calculate the coefficient of life for the vehicle. Using Equations 12.23, and 12.24 from Reference 1, pp 576, to calculate the coefficient of life and drag for the vehicle under supersonic conditions based on the angle of attack and Mach number.
Wave Drag
Also under the supersonic method is wave drag due to boundary layer thickness. The equations used are 92, 93 and 94 from Reference 4, with the sweep angle being the 90 - angle of attack and considering the whole vehicle as a single wing.
Viscous Interaction Effects
Finally there are the viscous interaction effects on the vehicle. The viscous interaction occurs when the boundary layer becomes thick due to the low densities at high altitudes. The effect is found by first calculation the viscous interaction coefficient, VI. The equation for VI can be found on page 41 in the methods hand out. Next VI is used to calculate the ratio of actual L/D and L/Dinviscid, using Equation 95 from Reference 4, on page 41. Finally, the CD,actual is found by using Equation 96 in the methods handout and by using the Cd calculated from Newtonian Theory. Cl is assumed to be unaffected by any viscous interaction effects for this method.
Combined Effects
In order to have a smooth transition from hypersonic to supersonic flow, at Mach 5, it is assumed the flow starts to be come supersonic, and becomes totally supersonic flow at Mach 4. In between Mach 5 and 4, a weighted average based on the Mach number is used to combine the two different flight conditions. Finally, the viscous interaction effect is added in for all three cases.
5.3.2 Aerothermodynamics
The next step was to calculate the heating values of the Mars entry vehicle. There were three different types of heating points considered, stagnation points, leading edge points and flat plate thermal points. The stagnation points were found first, then using the location of the stagnation points to calculate the heating values at the thermal points, then using both the stagnation and flat plate heating values to calculate the leading edge heating values. Currently there are two stagnation points, one on the front of the body at 45 from the x-axis and at the intersection of the fins and the body. The reason the 45 point is the stagnation point is because most of the time the vehicle is flying at that angle of attack and that would cause the stagnation point for the body to appear there. There are four flat plate thermal points along the body and one on the fin, and there are two leading edge points on the fin, the first having an arc length of rlead, and the second being on the tip of the fin. Figure 5.5 shows the locations of each of the thermal points on the vehicle.
Stagnation Points
The stagnation point heating values were found using Equation 47 Reference 4 and Equation 1 from Reference 5. Equation 47 is the stagnation heat-transfer rate to the vehicle were V is the free stream velocity, is the free stream density, rn is the nose radius of the vehicle or the radius of the leading edge of the fins depending on which stagnation point, and Cpw is the specific heat of the wall, and Tw is the wall temperature at the stagnation point. Equation 1 is the radiative heating at the stagnation point were C is a constant depending on the atmosphere, rn is the radius of the nose of the vehicle or the radius of the leading edge of the fins depending on which stagnation point, is the free stream density, a and b are constants depending on the atmosphere, and f(V) is a function depending on velocity. A curve fit was created from the values found in Table 1 from Reference 5. In the code the velocities used to in the f(V) calculation are limited to 6000-9000 m/s. If the velocity is larger then 9000, it is set to 9000, if it is smaller then 6000, it is set to 6000. The reason for doing this is because the peak radiative heating values occur within that velocity span so there is no need to calculate beyond those velocities. Once the two sets of values are calculated, they are added together to form a total value for the one stagnation point at the front of the vehicle, and for one at the stagnation point on the fins.
Flat Plate Thermal Points
Once the stagnation points have been found, the thermal points were found by using the flat-plate heat-transfer equations. The basic equation used to find the heating value at any given point is Equation 52 from Reference 4. The value is the free stream density and the V is the free stream velocity. The C is the heating parameter. C will vary depending on the conditions of the flow and the angle of attack of the vehicle. First the angle of attack is checked against the equation Msin found on page 30 in Reference 4. If the value for Msin is greater than 1, then the flow has large angles of attack and Tauber method is used. If the value for Msin is less then or equal to 1, then White’s method for small angles of attack is used.
For the two different methods, the location of transition is needed. Reusing Equations 71 and 72 from Reference 4 to get the transition location. For the large angle of attack with laminar flow, C is found from Equation 53 on page 30 of Reference 4. The equation has the variables , the angle of attack for this case, gw, the ratio of wall enthalpy to total enthalpy, found by equation 54, and x is the distance from the stagnation point. For the turbulent case, there are two different C values, one for a free stream velocity greater then 3962 m/s, Equation 56, and one for less then or equal to 3962 m/s, Equation 55. Once C is found, the heating rate at that point can be found.
For the small angle of attack case, there is also a laminar and turbulent case. The laminar case requires the calculation of the reference temperature from the free stream temperature, wall temperature and the free stream Mach number using equation 57. Next, use the reference temperature to calculate C* from equation 58. Then use Equation 59 to get the adiabatic wall temperature, and Equation 60 to get the skin-friction coefficient at the thermal point. In Equation 59, Pr is a constant equal to approximately 0.72. And in equation 60, Rexe is the Reynolds number calculated at the thermal point, using free stream conditions. Finally Equation 61 is used to get the heat-transfer coefficient from the skin-friction coefficient, which is then used with Equation 62 to get the heat transfer to the wall. For the turbulent case for a small angle of attack, the skin-friction coefficient is calculated using Equation 63, and the adiabatic wall temperature is calculated from Equation 64. The rest of the steps are the same as the laminar condition.
Leading Edge
Since the fins have the two leading edge points, a special heating equation is needed to calculate the heat flux, Equation 65 from Reference 4. The equation uses the sweep angle , and the heating values from the stagnation point and the flat plate heating value at that leading edge point to create a heating value for the leading edge.