**3. Modeling and analysis of flow-thermal coupling problem**

The coupling mechanism of fluid-thermal coupling problem (or fluid-solid conjugate heat transfer problem) is a physical process of interaction between aeroheating within the fluid and the heat transfer within the solid through the fluid-solid coupling interface. When the vehicle flies at hypersonic speed within the atmosphere, it will face strong aeroheating due to strong shock compression and viscous friction. A part of aerodynamic heat flux radiates from the surface while the remaining transfers to the internal structure. The heat into the solid structure is closely related to the structure layout, material properties and various heat transfer boundaries of radiation and convection within the solid, thus forming specific heat distribution characteristics within the solid structure in the form of transient temperature field distribution. Meanwhile, the heat distribution characteristics within the solid structure, in turn, also restricts aeroheating further entering the internal solid structure through the variation of the wall temperature. It represents a strong two-way coupling relationship between the aeroheating environment of the external flowfield and the thermal response of internal solid structure. The aeroheating environment as thermal load is active excitation, which changes continuously along the flight trajectory so that the fluid-thermal coupling problem appears as a sustained non-transient physical coupling process.

This coupling process involves three different time scales, that is, the characteristic time of the dynamic flight trajectory, the characteristic time of the flow response and the characteristic time of the structural thermal response. Research and development of the coupling analysis strategy should take full account of the obvious difference in the above time scales. Two concepts are introduced herein: (1) static flight trajectory, which refers to the flight state (height, speed and angle of attack) remaining constant over time; and (2) dynamic flight trajectory, which refers to the flight state (height, speed and angle of attack) changing continuously over time.

Dirichlet-Neumann model [7], which has been validated for the fluid-thermal coupling prob-

is firstly given to the solid structure, and then the temperature distribution of the solid structure on the fluid-solid coupling interface is transferred to the fluid domain as a Dirichlet boundary condition of the flowfield calculation by the interface information transfer

**2.** By calculating the steady flowfield within the fluid domain based on the imposed boundary condition of temperature, wall heat flux distribution of the steady flowfield can finally

**3.** The wall heat flux distribution of the steady flowfield is transferred to the solid domain as the Neumann boundary condition for the heat transfer calculation of the solid structure by the

> 0 to *t*<sup>0</sup>

based on the imposed heat flux boundary conditions to obtain the solid structure tempera-

domain as the Dirichlet boundary condition for the flowfield calculation by the interface

**6.** When the calculation in one coupling time step has been completed, the calculation will

, an initial constant temperature or temperature field distribution

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 + ∆*t*C within the solid domain can be done

 + ∆*t*C is transferred to the fluid

lem solved by partitioned coupling method.

0

interface information transfer method.

ture field distribution at *t*

information transfer method.

**4.** Calculation of transient heat transfer from *t*

0  + ∆*t*C.

**5.** The solid structure wall temperature distribution at time *t*<sup>0</sup>

continue in the next time step until all the time steps are covered.

**1.** At the initial time *t*

method.

be obtained.

The loosely-coupled analysis strategy can be summarized as follows:

**Figure 3.** Loosely-coupled analysis strategy for fluid-thermal coupling problems.

## **3.1. Coupling analysis strategies based on static flight trajectory**

#### *3.1.1. Loosely-coupled analysis strategy*

For hypersonic fluid-thermal coupling problems, in most cases, the characteristic time scale of structural thermal response is much longer than that of the flow response. Thus, the flow response can be assumed to be frozen over time compared with the structure thermal response. In other words, when the hypersonic flow is disturbed at certain time, a steady state can be reached instantaneously without relaxation process. Based on this physical assumption, the partitioned coupling method by calculating the steady flowfield and the transient structure heat transfer respectively is a good approximation, which can greatly improve the computational efficiency of coupling analysis.

A loosely-coupled analysis strategy for hypersonic fluid-thermal coupling problem based on static flight trajectory [6] is shown in **Figure 3**. ∆*t* F is the flowfield calculation time step, ∆*t* <sup>T</sup> is the structure heat transfer calculation time step. ∆*t* C is the coupling calculation time step, which can be set as several times of the structure heat transfer calculation time step, that is ∆*t* C = n ∙ ∆*t* <sup>T</sup> (*n* = 1, 2, 3, ⋯). The interface coupling relations are achieved by the Modeling and Analysis of Fluid-Thermal-Structure Coupling Problems for Hypersonic Vehicles http://dx.doi.org/10.5772/intechopen.70658 115

**Figure 3.** Loosely-coupled analysis strategy for fluid-thermal coupling problems.

**3. Modeling and analysis of flow-thermal coupling problem**

a sustained non-transient physical coupling process.

*3.1.1. Loosely-coupled analysis strategy*

114 Advances in Some Hypersonic Vehicles Technologies

computational efficiency of coupling analysis.

C = n ∙ ∆*t*

step, ∆*t*

step, that is ∆*t*

on static flight trajectory [6] is shown in **Figure 3**. ∆*t*

<sup>T</sup> is the structure heat transfer calculation time step. ∆*t*

The coupling mechanism of fluid-thermal coupling problem (or fluid-solid conjugate heat transfer problem) is a physical process of interaction between aeroheating within the fluid and the heat transfer within the solid through the fluid-solid coupling interface. When the vehicle flies at hypersonic speed within the atmosphere, it will face strong aeroheating due to strong shock compression and viscous friction. A part of aerodynamic heat flux radiates from the surface while the remaining transfers to the internal structure. The heat into the solid structure is closely related to the structure layout, material properties and various heat transfer boundaries of radiation and convection within the solid, thus forming specific heat distribution characteristics within the solid structure in the form of transient temperature field distribution. Meanwhile, the heat distribution characteristics within the solid structure, in turn, also restricts aeroheating further entering the internal solid structure through the variation of the wall temperature. It represents a strong two-way coupling relationship between the aeroheating environment of the external flowfield and the thermal response of internal solid structure. The aeroheating environment as thermal load is active excitation, which changes continuously along the flight trajectory so that the fluid-thermal coupling problem appears as

This coupling process involves three different time scales, that is, the characteristic time of the dynamic flight trajectory, the characteristic time of the flow response and the characteristic time of the structural thermal response. Research and development of the coupling analysis strategy should take full account of the obvious difference in the above time scales. Two concepts are introduced herein: (1) static flight trajectory, which refers to the flight state (height, speed and angle of attack) remaining constant over time; and (2) dynamic flight trajectory, which refers to

For hypersonic fluid-thermal coupling problems, in most cases, the characteristic time scale of structural thermal response is much longer than that of the flow response. Thus, the flow response can be assumed to be frozen over time compared with the structure thermal response. In other words, when the hypersonic flow is disturbed at certain time, a steady state can be reached instantaneously without relaxation process. Based on this physical assumption, the partitioned coupling method by calculating the steady flowfield and the transient structure heat transfer respectively is a good approximation, which can greatly improve the

A loosely-coupled analysis strategy for hypersonic fluid-thermal coupling problem based

time step, which can be set as several times of the structure heat transfer calculation time

F

<sup>T</sup> (*n* = 1, 2, 3, ⋯). The interface coupling relations are achieved by the

is the flowfield calculation time

<sup>C</sup> is the coupling calculation

the flight state (height, speed and angle of attack) changing continuously over time.

**3.1. Coupling analysis strategies based on static flight trajectory**

Dirichlet-Neumann model [7], which has been validated for the fluid-thermal coupling problem solved by partitioned coupling method.

The loosely-coupled analysis strategy can be summarized as follows:


#### *3.1.2. Tightly-coupled analysis strategy*

In the loosely-coupled analysis strategy, only one information exchanges between the fluid and the solid in each coupling calculation time step, and thus the coupling calculation is relatively efficient. However, this loosely-coupled analysis strategy does not strictly satisfy time synchronization of interface coupling relations, which reduces the accuracy of coupling calculation. On the basis of the above loosely-coupled analysis strategy, the sub-iteration strategy can be introduced to each coupling calculation time step, which allows multiple information exchanges between the fluid and the solid. The time marching calculations of flowfield and structure heat transfer are repeated until the convergence, and then calculation will continue in the next coupling time step, which forms a tightly-coupled analysis strategy. Obviously, the tightly-coupled analysis strategy improves the accuracy of the coupling calculation, but it requires additional sub-iteration calculation which decreases the calculation efficiency.

**Figure 4** shows the tightly-coupled analysis strategy of hypersonic fluid-thermal coupling problem based on static flight trajectory. The tightly-coupled strategy adds a step to repeat step (2) to step (5) until the convergence is satisfied.

#### **3.2. Coupling analysis strategy based on dynamic flight trajectory**

If the flight trajectory of the vehicle varies continuously over time, the influence of the flight trajectory variation should be considered in the coupling analysis. As the flight trajectory variation is the macroscopic motion of the vehicle as a rigid body and the time span of the continuous flight is large, the flight trajectory variation over time is relatively slow. Therefore, the time scale of affecting aeroheating characteristics is much larger than the coupling time scale between the flowfield and the heat transfer. A coupling analysis strategy for hypersonic fluid-thermal coupling problem based on dynamic flight trajectory is then shown in **Figure 5**.

It is to discretize the fluid-thermal coupling problem of the sustained dynamic flight trajectory into a set of fluid-thermal coupling problems of quasi-static flight trajectories in chrono-

**1.** The flight trajectory of the vehicle is regarded as a continuous flight state function in the

i

point is in which the internal flight conditions remain constant, set as the average of flight

pling problem based on the static flight trajectory. The appropriate coupling calculation

**3.** After the coupling calculation of one discrete flight state point, the coupling calculation of the next discrete flight state point in time order will be done until the end of the entire

The geometry of hypersonic vehicle model is shown in **Figure 6**. The head part is made of C/C composites, and the rest of fuselage is made of TB6 titanium alloy. The integrated analysis program platform Hypersonic Computational Coupling Dynamics (HyCCD) for

P of each discrete flight state, the problem is regarded as the fluid-thermal cou-

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conditions at the starting point and at the ending point of the discrete flight state.

<sup>P</sup> is selected to discretize the flight trajectory into

and the duration of each discrete flight state

logical order. The details are listed below:

**2.** During ∆*t*

flight.

time step ∆*t*

time domain. The appropriate time step ∆*t*

a series of discrete flight state point (*H*, *Ma*, *α*)

**Figure 5.** Coupling analysis strategy based on dynamic trajectory.

C = ∆*t*P/*n* (*n* = 1, 2, 3, ⋯) is selected.

**3.3. Fluid-thermal coupling analysis of hypersonic vehicle**

**Figure 4.** Tightly-coupled analysis strategy for fluid-thermal coupling problems.

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**Figure 5.** Coupling analysis strategy based on dynamic trajectory.

*3.1.2. Tightly-coupled analysis strategy*

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step (2) to step (5) until the convergence is satisfied.

**3.2. Coupling analysis strategy based on dynamic flight trajectory**

**Figure 4.** Tightly-coupled analysis strategy for fluid-thermal coupling problems.

In the loosely-coupled analysis strategy, only one information exchanges between the fluid and the solid in each coupling calculation time step, and thus the coupling calculation is relatively efficient. However, this loosely-coupled analysis strategy does not strictly satisfy time synchronization of interface coupling relations, which reduces the accuracy of coupling calculation. On the basis of the above loosely-coupled analysis strategy, the sub-iteration strategy can be introduced to each coupling calculation time step, which allows multiple information exchanges between the fluid and the solid. The time marching calculations of flowfield and structure heat transfer are repeated until the convergence, and then calculation will continue in the next coupling time step, which forms a tightly-coupled analysis strategy. Obviously, the tightly-coupled analysis strategy improves the accuracy of the coupling calculation, but it requires additional sub-iteration calculation which decreases the calculation efficiency.

**Figure 4** shows the tightly-coupled analysis strategy of hypersonic fluid-thermal coupling problem based on static flight trajectory. The tightly-coupled strategy adds a step to repeat

If the flight trajectory of the vehicle varies continuously over time, the influence of the flight trajectory variation should be considered in the coupling analysis. As the flight trajectory variation is the macroscopic motion of the vehicle as a rigid body and the time span of the continuous flight is large, the flight trajectory variation over time is relatively slow. Therefore, the time scale of affecting aeroheating characteristics is much larger than the coupling time scale between the flowfield and the heat transfer. A coupling analysis strategy for hypersonic fluid-thermal coupling problem based on dynamic flight trajectory is then shown in **Figure 5**.

It is to discretize the fluid-thermal coupling problem of the sustained dynamic flight trajectory into a set of fluid-thermal coupling problems of quasi-static flight trajectories in chronological order. The details are listed below:


#### **3.3. Fluid-thermal coupling analysis of hypersonic vehicle**

The geometry of hypersonic vehicle model is shown in **Figure 6**. The head part is made of C/C composites, and the rest of fuselage is made of TB6 titanium alloy. The integrated analysis program platform Hypersonic Computational Coupling Dynamics (HyCCD) for

**Figure 8** shows the wall temperature at the leading edge stagnation point of the symmetric plane along the dynamic trajectory. The wall temperature evolution without considering the wall radiation effect is also given in the figure for comparison. It can be seen that: (1) from the starting point *t* = 0 s to the cruise state *t* = 50 s, the wall temperature of the structure gradually increases while the aeroheating capacity of the external flowfield rises gradually due to the increasing flight speed, and thus more and more aerodynamic heat is transferred into the internal solid structure through the interface. The solid structure keeps storing heat; (2) during the cruise *t* = 50–150 s, although the flight conditions no longer change and the aeroheating capacity of the external flowfield remains constant, aerodynamic heat is still continuously transferred into the internal structure due to the fluid-solid heat transfer coupling effect. The solid structure still keeps storing heat. Meanwhile, as the wall temperature continues to rise, less and less aerodynamic heat will be transferred into the internal structure. If the cruise flight time is long enough, the fluid-solid heat transfer coupling will eventually reach equilibrium; and (3) in descending *t* = 150–210 s, the flight speed begins to decrease, the aeroheating capacity begins to weaken while the wall temperature has been extremely high, reaching the temperature peak along the entire trajectory. The solid struc-

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ture begins to release heat, and thus the wall temperature decreases gradually.

**Figure 8.** The wall temperature at stagnation point of the symmetric plane leading edge.

**Figure 9** show the temperature distribution of I-I cross-section (see **Figure 8**) within the solid structure, intuitively presenting how the temperature distribution within the structure changes with the dynamic trajectory. It should also be noted that the heat distribution and transmission within the structure will change with the trajectory of the vehicle, thus exhibiting dynamic multi-field coupling spatial-temporal characteristics. It is an effective

**Figure 6.** The geometric shape and CFD/FEM computational grid.

hypersonic fluid-thermal coupling problem is realized by integrating hypersonic aerothermodynamic numerical simulation program Hypersonic Computational Fluid Dynamics (HyCFD) and the finite element heat transfer analysis software (ANSYS Mechanical APDL). The external CFD computational grid is multi-block structured grid, the internal FEM grid is unstructured grid.

The actual flight of hypersonic vehicles usually includes climbing, cruise and descending. A simple flight trajectory is assumed here, as is shown in **Figure 7**. The flight time along the trajectory is 210 s, 0–50 s for climbing, 50–150 s for cruise and 150–210 s for descending. The discrete time step of the selected flight trajectory is ∆*t* P = 5 s, and the flight trajectory is divided into 42 discrete flight states. The loosely-coupled analysis strategy is used for the fluid-thermal coupling analysis, and the fluid-solid coupling time step is ∆*t* C = 5 s. The chemical non-equilibrium gas model is used for flowfield calculation. Non-catalytic wall is selected as the boundary condition. The initial temperature is 300 K, the inner wall temperature keeps at 300 K during the flight, and the outer wall emissivity is *ε* = 0.8.

**Figure 7.** The dynamic flight trajectory and its discretization.

**Figure 8** shows the wall temperature at the leading edge stagnation point of the symmetric plane along the dynamic trajectory. The wall temperature evolution without considering the wall radiation effect is also given in the figure for comparison. It can be seen that: (1) from the starting point *t* = 0 s to the cruise state *t* = 50 s, the wall temperature of the structure gradually increases while the aeroheating capacity of the external flowfield rises gradually due to the increasing flight speed, and thus more and more aerodynamic heat is transferred into the internal solid structure through the interface. The solid structure keeps storing heat; (2) during the cruise *t* = 50–150 s, although the flight conditions no longer change and the aeroheating capacity of the external flowfield remains constant, aerodynamic heat is still continuously transferred into the internal structure due to the fluid-solid heat transfer coupling effect. The solid structure still keeps storing heat. Meanwhile, as the wall temperature continues to rise, less and less aerodynamic heat will be transferred into the internal structure. If the cruise flight time is long enough, the fluid-solid heat transfer coupling will eventually reach equilibrium; and (3) in descending *t* = 150–210 s, the flight speed begins to decrease, the aeroheating capacity begins to weaken while the wall temperature has been extremely high, reaching the temperature peak along the entire trajectory. The solid structure begins to release heat, and thus the wall temperature decreases gradually.

**Figure 9** show the temperature distribution of I-I cross-section (see **Figure 8**) within the solid structure, intuitively presenting how the temperature distribution within the structure changes with the dynamic trajectory. It should also be noted that the heat distribution and transmission within the structure will change with the trajectory of the vehicle, thus exhibiting dynamic multi-field coupling spatial-temporal characteristics. It is an effective

hypersonic fluid-thermal coupling problem is realized by integrating hypersonic aerothermodynamic numerical simulation program Hypersonic Computational Fluid Dynamics (HyCFD) and the finite element heat transfer analysis software (ANSYS Mechanical APDL). The external CFD computational grid is multi-block structured grid, the internal FEM grid

The actual flight of hypersonic vehicles usually includes climbing, cruise and descending. A simple flight trajectory is assumed here, as is shown in **Figure 7**. The flight time along the trajectory is 210 s, 0–50 s for climbing, 50–150 s for cruise and 150–210 s for descending.

divided into 42 discrete flight states. The loosely-coupled analysis strategy is used for the

cal non-equilibrium gas model is used for flowfield calculation. Non-catalytic wall is selected as the boundary condition. The initial temperature is 300 K, the inner wall temperature keeps

<sup>P</sup> = 5 s, and the flight trajectory is

C = 5 s. The chemi-

The discrete time step of the selected flight trajectory is ∆*t*

**Figure 6.** The geometric shape and CFD/FEM computational grid.

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at 300 K during the flight, and the outer wall emissivity is *ε* = 0.8.

**Figure 7.** The dynamic flight trajectory and its discretization.

fluid-thermal coupling analysis, and the fluid-solid coupling time step is ∆*t*

is unstructured grid.

**Figure 8.** The wall temperature at stagnation point of the symmetric plane leading edge.

HyCCD can effectively predict and analyze the thermal response characteristics and principals of the solid rigid structures inside the hypersonic vehicle under sustained flight conditions. The accurate prediction and analysis will be an effective way to optimize thermal protection design and to study new concepts and methods of thermal protection system of hypersonic vehicles.

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**4. Modeling and analysis of fluid-thermal-structural coupling problem**

tural deformation.

**Figure 10.** Fluid-thermal-structural coupling model.

During the sustained flight of an actual hypersonic vehicle within the atmosphere, the strong two-way coupling of aerothermal environment with thermal response within solid structures causes the heat distribution in the form of transient temperature field. It not only has dynamic effects on the properties of solid structures also causes thermal stress due to the temperature gradient. Meanwhile, the resultant thermal strain from the thermal stress influences the heat distribution through the deformation of the solid structures. The large structural deformation even affects the aerodynamic forces/heat in the external flowfield. On the other hand, the aerodynamic forces lead to the structural stress and structural within solid structures, which can also affect aerodynamic forces/heat in the external flowfield in the form of struc-

The fluid-thermal-structural coupling model is shown in **Figure 10**. The volumetric coupling of the aerodynamic forces and aerodynamic heat of the flowfield within the fluid is described by unified fluid governing equations, which is solved by computational fluid dynamics (CFD) to obtain the parameters of aerodynamic forces/heat. The thermal load (wall heat flux *q*) and force load (wall pressure *p*) are imposed on the solid through the fluid-solid coupling interface. Within the solid, the thermal response is described by governing equations of heat conduction, while the stress/strain are described by governing equations of thermoelastics.

**Figure 9.** The temperature distribution of the I-I section plane within the solid structure. (a) *t* = 25 s, (b) *t* = 55 s, (c) *t* = 100 s, (d) *t* = 150 s, (e) *t* = 180 s, and (f) *t* = 210 s.

to optimize the integrated design of thermal protection and to study new concepts and methods of thermal protection by accurately predicting and analyzing the temporal and spatial distribution characteristics and the transmission of heat within the vehicle solid structure for guidance.

The preliminary analysis and research on coupling characteristic and influencing factors of the hypersonic fluid-thermal coupling problem reveal the spatial-temporal distribution characteristics of the fluid-solid heat transfer coupling and the influence of wall radiation effect on the sustained flight conditions. There is a close coupling between the aeroheating of the flowfield and the heat transfer of the structure. The heat distribution and transfer within the structure change with the dynamic trajectory, which displays the spatial and temporal characteristics of multi-physics coupling. The integrated analysis method and the program platform HyCCD can effectively predict and analyze the thermal response characteristics and principals of the solid rigid structures inside the hypersonic vehicle under sustained flight conditions. The accurate prediction and analysis will be an effective way to optimize thermal protection design and to study new concepts and methods of thermal protection system of hypersonic vehicles.
