Pseudo-transient 3D Conjugate Heat Transfer Simulation and Lifetime Prediction of a Rocket Combustion Chamber

Due to symmetry, the CHT model includes half of the fatigue specimen, eight and a half cooling channel flows and a quarter of the hot gas flow. For the coolant flow simulation the Finite Volume Method (FVM) based open source Computational Fluid Dynamics (CFD) tool OpenFOAM is utilized. On structural side the Finite Element Method (FEM) based commercial Computational Solid Mechanics (CSM) tool Abaqus is used. For the hot gas flow simulation the FVM based commercial CFD tool Ansys Fluent is applied. For controlling the field solvers and managing the data exchange between the 11 domains the in-house tool ifls [18] together with the Dirichlet–Neumann coupling scheme is utilized. On the coupling surface heat fluxes, calculated in the fluid domain boundaries, are applied as boundary condition for the solid domain. Vice versa, temperatures, calculated in the solid domain boundaries, are utilized as boundary condition for the fluid domains. The coupling is implicit, therefore the interface quantities will be exchanged during one time step until convergence. Aitken’s dynamic relaxation is applied to accelerate the convergence of the equilibrium iteration. It is assumed that the time scales of the fluid domains are much smaller than the time scales of the structure domain. Therefore, the structure domain is solved transiently, whereas the fluid domains are solved under the assumption of steady-state conditions. The reader is referred to [12] for a detailed description of the governing equations and the material models used for the different domains.

For the simulation of the coolant domain the OpenFOAM steady-state solver buoyantSimpleFoam with the BSL-EARSM turbulence model [17] is utilized. The material data are considered to be temperature dependent. The turbulent Prandtl number is assumed as 0.85. Figure 2a shows the computational domain of the half of the central cooling channel. Due to simplicity the original cross section of the inlet and outlet channel was modified from an elliptical to a rectangular cross section without changing the hydraulic diameter. Table 1 shows the boundary conditions. n indicates the normal vector of the surfaces. \(U_n\) and \(U_p\) are the velocity components in direction n and in plane direction. T is the temperature, p the pressure, \(\omega \) the turbulence specific dissipation, k the turbulence kinetic energy, \(\nu _t\) the turbulent kinematic viscosity, \(\alpha _t\) the turbulent thermal diffusivity, \(\nu \) the kinematic viscosity and y the wall distance of the cell center. The temperature of the walls is calculated in the structure domain and substituted iteratively. All other cooling channel domains are consisting of this half cooling channel model and its counterpart mirrored with respect to the symmetry plane. A computational grid with 500,296 block-structured cells and a dimensionless wall distance \(y^+\) of in average 0.32 is used for the central cooling channel. Each grid of the eight other cooling channels consists of 1,000,592 block-structured cells per channel.

For simulating the hot gas domain Fluent’s steady state pressure-based solver is utilized. For modeling the Reynolds stresses the standard \(k-\epsilon \)-model by Jones and Launder [15] with a two-layer approach in the near-wall region (see [1]) is applied. A non-adiabatic steady diffusion Flamelet model is used in a preprocessing step to calculate the relationships of the instantaneous temperature, density and species mass fractions to the total enthalpy, mixture fraction and its variance. The kinetic chemistry scheme of [21] is applied, containing 21 species and 97 reactions. A \(\beta \)-probability density function is assumed, to describe the turbulent, temporal fluctuations of the mixture fraction. Figure 2b shows the hot gas domain and Table 2 its boundary conditions for both pressure stages. \(\epsilon \) is the rate of dissipation of turbulence energy, \(\overline{f}\) is the Favre mean mixture fraction and \(\overline{f'^2}\) its variance. The temperature of the specimen is calculated in the structure domain and substituted iteratively. In Table 2 WF means that the turbulence quantities at the walls are calculated by the Ansys Fluent Advanced Wall Treatment (see [1] for details). A computational grid with \(1.39\cdot 10^6\) block-structured cells and a dimensionless wall distance \(y^+\) of in average 0.83 is used.

Abaqus Standard is applied to solve the transient heat conduction inside the structure. The temperature dependence of the material data for CuCr1Zr is considered. Figure 3 shows the boundary conditions of the structural domain. The coupling surface to the hot gas and coolant are colored red and blue. All other surfaces, including the green symmetry plane are assumed as adiabatic. Equivalent to the coolant simulation, but inconsistent with the experiment, the inlet and outlet channel’s cross sections are rectangular. In addition, the notches and the holes for the screws and thermocouples are neglected. The mesh consists of 372,670 nodes and 322,266 linear brick elements.

Figure 4a shows the temperature transients for thermocouple T6 (see Fig. 1b) and its equivalent nodes for two simulation cases. After ignition at t = 0 s the temperature increases with a decreasing gradient. At t = 7.5 s the second load stage starts, which increases the temperature gradient rapidly. After that the gradient decreases again until t = 27.5 s, where the flames are extinguished and the maximum temperatures are reached. The temperatures decrease rapidly after extinguishing.

In both simulation cases the turbulent Prandtl number \(Pr_t\) of the hot gas domain is varied between 0.7 and 0.9. A lower \(Pr_t\) increases the wall heat flux. For \(Pr_t = 0.7\) the transient behaviour agrees very well with the experiment. The maximum deviation of 34 K during the hot run is reached after 10.4 s. After that the deviation is decreasing. During the post-cooling phase the deviation is increasing again up to 69 K. Therefore, it can be concluded, that the heat fluxes of the coolant and hot gas domain are slightly underestimated.

Figure 4a shows the axial temperature distribution at the end of the hot gas run (t \(=\) 27.5 s) for experiment and simulation (\(Pr_t=0.7\)). The red curves show the temperature distribution in 3 mm distance to the hot gas exposed surface and the blue curves in 5 mm distance, respectively. The temperature is decreasing in x-direction (in hot gas flow direction). This is mostly caused by the high heat flux of the bumping coolant in the cooling channel inlet region (counterflow cooling).

The temperature distribution shows a good agreement between simulation and experiment. The vertical temperature difference between 3 and 5 mm distance to the hot gas exposed surface is around 50 K in the experiment and around 45 K in the simulation. This also indicates that the heat flux is slightly underestimated in the fluid domains. The axial temperature gradient agrees well.

The thermal boundary conditions at the cooling channels and the hot gas wall for one cycle with a total duration of 60 s are summarized in Table 3. Therein, \(\alpha _{hg}\), \(\alpha _{cf}\), \(T_{hg}\), and \(T_{cf}\) denote the convective heat transfer coefficients and the bulk temperatures of the hot gas (hg) side and the cooling fluid (cf), respectively. Figure 6 shows the temperature distribution within the specimen reaching a maximum of 1113.18 K at the hot gas wall after the second hot run phase (i.e. 29.5 s). Furthermore, the temperature evolution over time at three different positions which are close to the region of interest where failure is expected to occur is illustrated and compared to experimental results coming from the thermocouple measurements T17, T18 and T20 (cf. Fig. 1). It can be observed that the numerical and experimental results agree very well, especially close to the symmetry plane (i.e. T17 and T18). For T20 a slight deviation can be recognized where higher temperatures are reached in the simulation, especially after the two hot run phases. This stems from the fact that due to simplicity and the lack of experimental data, the hot gas film coefficient \(\alpha _\mathrm{hg}\) was chosen to be constant during each hot run phase (cf. Table 3). This, however, is not of major concern, since final failure is expected to occur close to the symmetry plane which will be further investigated in Sect. 3.2.

The lifetime of the fatigue specimen which is made of a copper alloy (CuCr1Zr) is limited by the failure of the cooling channels. Therefore the material and damage modeling of CuCr1Zr is of major concern. To this end, a viscoplastic damage model presented in former works (see e.g. Kowollik et al. [16] or Fassin et al. [10]) is utilized which accounts for nonlinear kinematic and isotropic hardening, respectively. Furthermore, rate-dependence of Perzyna type and Lemaitre type ductile damage are incorporated into the model. The applied material parameters are chosen as in Fassin et al. [9] and Barfusz et al. [3]. Using a staggered simulation scheme, the thermomechanical coupling is performed only in one way considering the influence of the temperature on the mechanical behavior, but not vice versa. Temperature dependence of the mechanical analysis is taken into account by thermal expansion and the temperature dependence of the material parameters. The temperature field resulting from the thermal analysis (see Sect. 3.1) serves as an input for the quasi-static simulation consisting of multiple cycles.

The mechanical boundary conditions at the cooling channels and the hot gas wall are given in Table 4. It contains the respective pressures \(p_{hg}\) and \(p_{cf}\). Efficient and robust low-order continuum finite elements based on reduced integration with hourglass stabilization¹ are used for the spatial discretization. Due to computational efficiency, only one element is utilized in depth direction whereby the displacement in this direction is fixed, leading to a plane strain state. Since the simulation is aborted at the beginning of the 48th cycle (no convergence), the following results are only shown up to the 47th cycle.

Figure 7 illustrates the contour of the scalar damage variable D after 47 cycles, ranging from the undamaged state \((D=0)\) to the fully damaged state \((D=1)\), the latter of which corresponds to macroscopic failure. It can be observed that the wall between the first two cooling channels next to the symmetry plane and the hot gas side represents the location of maximum damage accumulation (i.e. material degradation). Zooming into the region of interest after several cycles, the degradation process can be well monitored. Starting at the cooling side edge (cycle 42), damage grows diagonally through the wall (cycle 43). Afterwards, damage starts to proceed also from the opposite hot gas side center (cycle 44) and merges eventually into a diagonal macroscopic crack (cycle 47). Additionally, Fig. 8 gives the reader an impression of the accumulation of damage during the simulation of one and 47 cycles within three different elements² (i.e. cooling side edge, cooling side center, as well as hot gas side center). It is evident that damage increases most within the phase transitions of each cycle, since it is driven by the temperature gradient as well as the pressure difference between the coolant and the hot gas. This is why the maximum increase in damage takes place between the second hot run and the post cooling. During this transition phase, the greatest temperature and pressure differences occur. Furthermore, at this point the material properties are also the weakest here. Interestingly, when looking further at the development of damage over several cycles within the three considered elements (Fig. 8), it can be observed that the increase in damage is initially almost linear and then becomes highly non-linear. Particularly at the hot gas side center (red curve), an enormous increase in damage is observed after approx. 40 cycles, which leads to the crack spreading from both sides as already mentioned above.

In order to validate the numerical results from the quasi-static mechanical analysis also quantitatively, a comparison to experimental observations is made in Fig. 9. The cut view at the crack tip of the fatigue experiment after 48 cycles has been taken from the test campaign conducted by Hötte et al. [11]. After the 48th cycle, the breakthrough of the cooling channel (macroscopic failure) was first observed in the experiment. The maximum deformation of the specimen, which consists of bulging and thinning of the hot gas wall, takes place at the cooling channel next to the symmetry plane. Whereas the bulging of the hot gas wall is accurately represented by the simulation, the thinning process can be interpreted in the following way. Since the damage variable reaches the value of \(D=1\) within the diagonal process zone between the cooling side edge and the hot gas center, the elements above this region can no longer contribute to the load-bearing capacity of the specimen. Therefore, the elements above this process zone could also be thought of as being removed, which can ultimately be interpreted as thinning of the specimen. Since the deformed shape of the cooling channel wall after failure resembles the shape of a doghouse, the failure mode is frequently called doghouse effect in literature (see e.g. Riccius et al. [20]). In total, it can be stated that the doghouse effect can be well represented by the shown thermomechanical analysis scheme utilizing a viscoplastic damage model.