A Novel Optimization Strategy for Reducing the Initial Error of a Quasi-Steady Algorithm for Conjugate Heat Transfer

Abstract

The present study proposes a novel optimization strategy (NOS) for quasi-steady algorithms to optimize the initial error in the fast calculation of conjugate heat transfer (CHT) simulations. In this approach, the change in Nusselt number at the fluid–solid coupling interface is dynamically monitored, and the update of the flow field is turned off according to a given Nusselt variation standard to speed up the solution of the transient temperature field. The NOS has been applied to problems of convective heat transfer in solid parts with internal heat sources. The feasibility of NOS is first verified by using an undisturbed boundary example, and the results show that the optimization strategy reduces the initial error by 92.3% compared with the quasi-steady algorithm, and the calculation time is reduced by 50% compared with the traditional coupling algorithm. The NOS is then combined with the quasi-steady algorithm, and boundary transient disturbances are added to the case. The results indicate that the computational time for NOS and the quasi-steady algorithm is 2.6 and 2.9 times greater than that of traditional algorithms. Nevertheless, NOS significantly optimizes the relative error of the quasi-steady algorithm by 97.3% during the initial computation phase.

1. Introduction

Conjugate heat transfer phenomena are abundant in science and engineering fields [1], such as spacecraft thermal protection [2], gas turbine blade cooling [3,4], biomedicine [5], chemical processing [6], and heat dissipation in electronic devices [7]. The reliability of airborne radar equipment determines the survivability of aircraft in wartime, but the performance and reliability of radar equipment have a strong dependence on temperature and environmental conditions. Many flight conditions with severe thermal loads are relatively short, so to make radar equipment work reliably and stably, it is necessary to calculate the transient temperature distribution under dynamic flight conditions [8].

The conjugate heat transfer problem can be solved by theoretical analysis or numerical simulation, in addition to experimental studies. Perelman [9] first proposed the concept of conjugate heat transfer, and many scholars applied the conjugate heat transfer method to the thermal analysis of turbine blades [10,11,12,13]. Unsteady CHT is not limited to turbine blade cooling applications. It can also be found in modeling heating, cooling, and ventilation flow in building simulation [14,15,16]. Transient conjugate heat transfer analysis methods can be divided into fully coupled methods and loosely coupled methods. Both methods face the problem of high computational complexity in the transient process. In a fully coupled model, the ensemble solver requires the fluid and solid domains to have the same timescale. Solid conduction and fluid convection are solved simultaneously at each time step [17,18]. However, solid conduction and fluid convection differ significantly in timescales [12]. To ensure the accuracy and stability of the fluid solver, a small time step is usually chosen. In the process of conjugate heat transfer, the number of meshes is dominated by the fluid part, but a large number of fluid meshes is usually required to cause the time cost of the entire calculation process to be too large [19]. The authors of [20] adopted a method of scaling the thermal conductivity of solids to reduce the timescale of heat conduction. The calculation result of the fluid with a small step size is used as the solid thermal boundary at a large step size, but this method is only suitable for short-term conjugate heat transfer problems.

In the loosely coupled model, the partition solver is used to solve the fluid domain and the solid domain separately [21], the subdomains are associated through the fluid–solid interface, which usually requires additional algorithms for data transfer [22], and the calculation cost is also relatively high. However, the partition solver provides an asynchronous time-advancing method, and other algorithms have been proposed to improve computational efficiency [23,24,25,26].

The flow field is the most computationally expensive part of the solution. The above algorithm calculates the flow field continuously, and there is still the possibility of optimizing the speed of the calculation. Quasi-steady assumptions can be adopted for long-term transient conjugate heat transfer problems [27,28,29,30,31]. Fluid convection is regarded as a steady state, and solid conduction is regarded as a transient process. It iterates over the thermal conditions at the fluid–solid interface until equilibrium is reached, then solves for the next time step. The quasi-steady algorithm has been applied to coupled thermal simulations such as turbine thermal analysis [28], plate heat transfer under thermal conditions of solid rocket motors [29,30], and forced convection heat transfer in tubes [31]. The above application results show that the computational cost can be reduced significantly and the overall accuracy is maintained to a certain extent, but a large initial error is generated due to the use of a steady-state flow field at the initial moment. According to the above study, it can be found that although the quasi-steady algorithm can greatly reduce the computational speed for the long-term transient conjugation problem, the existence of initial errors has an important impact on the heat-sensitive devices, so this paper proposes a novel strategy about optimizing the quasi-steady initial errors.

In this paper, the steady-state calculation in the initial stage of the quasi-steady algorithm is replaced by the transient coupling calculation. A criterion that characterizes the effect of flow field changes on the temperature field is determined, and this criterion is used as the basis for judging the end of the transient coupling calculation. The feasibility of the optimization strategy is verified by two-dimensional forced convection heat transfer simulation without boundary disturbance, then the optimization strategy is combined with the quasi-steady algorithm and applied to the simulation with the addition of boundary disturbance, and the results show that the strategy maintains a similar computational speed as the quasi-steady algorithm while significantly optimizing the initial error.

This paper is constructed as follows. In Section 2, the development of the unsteady coupled CHT algorithm is completed based on OpenFOAM, and the accuracy of the algorithm is verified by experimental data. In Section 3, a novel optimization strategy is proposed and applied to the forced convection heat transfer simulation without boundary disturbance to verify the feasibility of the optimization strategy. In Section 4, the optimization strategy is combined with the quasi-steady-state algorithm, and the optimization effect of the strategy is analyzed in simulations with boundary disturbances added. In Section 5, research limitations are added to show where the strategy still needs to be improved in the future. The main conclusions are summarized in Section 6.

2. Numerical Methods

2.1. Baseline Solver for CHT

In this work, the unsteady incompressible Navier-Stokes equations are adopted for the fluid domain, and the governing equations are as follows:

Continuity equation:

$$\frac{\partial u_i}{\partial x_i}=0$$

(1)

Momentum equation:

$$\frac{\partial u_i}{\partial t}+u_j\frac{\partial u_i}{\partial x_j}=-\frac{\partial p}{\partial x_i}+v\frac{{\partial }^2{u}_i}{\partial x_j^2}$$

(2)

Energy equation:

$$\frac{\partial {\rho }_f{c}_{p,f}T}{\partial t}+{\rho }_f{c}_{p,f}{u}_j\frac{\partial T}{\partial x_j}=k_f\frac{{\partial }^{2}T}{\partial x_j^2}$$

(3)

For the solid domain, the velocity is zero, all the transport equations are no longer applicable, and the energy equation is transformed into an unsteady conduction equation:

$$\frac{\partial {\rho }_s{c}_{p,f}T}{\partial t}=k_s\frac{{\partial }^{2}T}{\partial x_j^2}+Q_s$$

(4)

The energy conservation is satisfied at the fluid–solid interface [9], expressed as

$$q_s\left(T_s\right)=-q_f\left(T_f\right)$$

(5)

$$T_s=T_f$$

(6)

Equation (5) represents the temperature and heat flow continuity on the interface’s solid (s) and fluid (f) sides.

The numerical method used in this paper is briefly described below. All analyses in this article are developed and implemented based on OpenFOAM.

2.2. Solver Validation

The test example is the forced convection cooling problem that flows through three rectangular heated solids between two parallel plates. This problem was first proposed by Davalath and Bayazitoglu [32] to simulate the cooling of integrated circuit components. The computational domain and boundary conditions of the problem are shown in Figure 1; the upper and lower walls are adiabatic, the inlet velocity is parabolic, the inlet temperature is 0, and the outlet pressure is 0. Monitoring points (a–n) are set around the three rectangular blocks. The fluid enters from the left side and flows out from the right side with a fully developed velocity profile. The heat generated in the three solid blocks can be assumed to have a uniformly distributed heat source term.

In this paper, we assume that the solid heat source Qs is 8, the thermal conductivity of the solid is 10 times that of the fluid, the kinematic viscosity of the fluid is 0.01408, and the Pr is 0.71. For more detailed parameters, please refer to [33].

In practical engineering calculations, unstructured meshes are highly adaptable to simulate complex geometries more finely and can dynamically assign grid cells and reduce the number of grid cells. To illustrate the adaptation of this novel optimization strategy to unstructured meshes, unstructured meshes are, therefore, used to validate the simulation results. Four different grid numbers were used to obtain the flow field parameters for grid-independent verification and to determine the appropriate number of grids. Figure 2 shows the variation in the temperature of the monitoring points in the grid with the grid number for different Reynolds numbers, and it can be seen that the grid containing 7984 linear triangular elements already has sufficient accuracy to characterize the flow field information, so this grid is used for subsequent calculations, as shown in Figure 3.

Figure 4 shows the temperature distribution of different monitoring points set at the fluid–solid interface. It can be seen that the calculation results are consistent with the literature.

3. New Optimization Strategy

3.1. Description of the Optimization Strategy

The traditional quasi-steady algorithm solves the steady flow field in the fluid region independently at the initial time. The fluid–solid coupling wall is set as a fixed temperature boundary, and the steady momentum, turbulence, and energy equations in the fluid region are solved simultaneously to obtain the steady flow field. Then, the transient heat transfer calculation is carried out for the fluid and solid regions at the same time, the fluid–solid coupling wall is set as the heat transfer coupling boundary, and the transient energy equation of the fluid and solid regions is solved simultaneously to obtain the temperature field distribution at each time until the next update time of the flow field.

The quasi-steady algorithm decomposes the complete transient calculation process into a series of steady-state couplings of the solid transient temperature response and the fluid at a specific time step. Since the steady-state solution of the fluid in the calculation of the initial state differs significantly from the actual initial flow field, large errors can occur in the initial calculation. To compensate for the initial calculation error, a tightly coupled calculation can be used for a specific initial state process, but it is crucial to determine the moment when the initial state ends. In fluid–solid heat transfer, the Nusselt number can be used to characterize the relative size of convective heat transfer and heat conduction. The average Nusselt number is solved using the following equation:

$$h_i=\frac{q_f{}_i}{T_w{}_i-T_{in}}$$

(7)

$$N{u}_i=\frac{h_{i}L}{k_f}$$

(8)

$$\overline{Nu}=\frac{\sum N{u}_i}{n}$$

(9)

where $i$ represents a certain solid boundary unit and $n$ is the total number of solid boundary units.

When the fluid tends to a stable state, the Nusselt number will also gradually stabilize. Therefore, this paper monitors the change in the average Nusselt number (DeltNu) on the solid surface and determines the end time (t_criterion) of the initial state when the change is less than the specified criterion, as shown in Figure 5. The flow field information at this time is used to update the temperature field in the fluid and solid the next time.

$$DeltNu=\left|\overline{N{u}_{pre}}-\overline{N{u}_{now}}\right|$$

(10)

3.2. Validation of the Optimization Strategy

The same numerical example as in Section 1 is used to verify the optimization strategy, where the inlet boundary is undisturbed, and the physical simulation time is 100 s. The quasi-steady algorithm updates the steady-state flow field only at the initial moment, and the three DeltNu change standards are set in the new optimization strategy, respectively 0.1, 0.01, and 0.001, which determine the end time of the coupling calculation under different standards. By comparing the calculation results of traditional coupling (Normal), quasi-steady (Quasi Steady), and new optimization strategy (NOS), the degree of optimization of the initial error under different standards is analyzed. The three algorithms use the same convergence criterion for pressure, temperature, and velocity of 1 × 10⁻⁷.

The variation curves of the average Nu on the surface of solids 1, 2, and 3 (S1, S2, S3) are shown in Figure 6. It is found that Nu gradually decreases with the heat transfer process and finally remains stable, indicating that the average temperature response of the solid surface gradually moves away from the inlet fluid temperature. The three solids using the quasi-steady algorithm produced large errors of about 5% in initial calculations. At the end of the final calculation, the error of Nu close to the solid at the entrance using the quasi-steady algorithm is close to 0 compared with the traditional algorithm, and the closer the solid position is to the entrance, the greater the relative error. The maximum relative error of the quasi-steady algorithm at the initial stage of calculation is about 7.6%, and the maximum error at the end of the calculation is about −6.5%.

The average temperature variation curve of the solid surface is shown in Figure 7. The maximum relative error arises at the initial moment of the calculation, which is about −22.1%. The relative errors of S1, S2, and S3 gradually decrease and remain stable at the final moment, and are 0.08%, 2%, and 6.7%, respectively. These results coincide with the variation in the average Nu on the solid surface.

With the new operation strategy, the maximum relative error of Nu during the calculation decreases with the decrease in the variation criterion, as shown in Figure 8. When the change standard is 0.1 and 0.001, the maximum relative errors of Nu in the three solids are 6% and −4%, respectively. There are still large deviations because the flow field is not sufficiently updated. However, when the standard is 0.001, the relative error of the whole calculation process is close to 0%, which is because when a small variation standard is used, the flow field has sufficient time to be updated and the transient change in the flow field produces less fluctuation on the coupled heat exchange between the flow and the solid, which also further illustrates the reasonableness of choosing this variation standard. The relative error of the average temperature of the solid surface is shown in Figure 9. At the variation standard of 0.1, 0.01, and 0.001, the maximum relative errors of the solid temperature are −9%, −5.5%, and −2%, respectively, and the relative errors of the calculated results become smaller as the variation standard is gradually reduced. When the criterion is 0.001, the maximum relative error of the quasi-steady algorithm is optimized to 92.3%.

The time-consuming comparison of three different algorithms under the same physical simulation time is counted, as shown in

Table 1. Computational time-consuming comparison of three different algorithms.

Algorithm	Nu_Criterion	t_criterion (s)	Computational Time (s)
Normal	-	-	218
Quasi Steady	-	-	41
NOS	0.1	0.96834	43
NOS	0.01	2.84694	66
NOS	0.001	11.2444	99

. From the t_criterion in the table, it can be found that the smaller the Nu_Criterion adopted by the new optimization strategy, the longer the time from the initial moment to the end of the coupling calculation. Therefore, the calculation time consumed is longer. The speed of the quasi-steady algorithm is about 5 times that of the traditional algorithm and has the largest calculation speed. When the standard of NOS is 0.1, the calculation time is close to quasi-steady. As the standard decreases, the time consumption gradually increases. When the standard is 0.001, the calculation speed is 2.2 times that of the traditional coupling algorithm. Although it is slower than the quasi-steady algorithm, the result has the highest calculation accuracy and the largest relative error is −1.7%.

4. Test Case and Results

It can be found from the verification examples that NOS optimizes the calculation error of the quasi-steady algorithm while increasing the speed of the traditional algorithm. Therefore, the velocity disturbance is added to the inlet boundary in the calculation example, and then the NOS is combined with the quasi-steady algorithm to update the flow field every time the velocity disturbance occurs.

The calculation process is divided into five time periods.

Table 2. Calculation time consumption for each time period.

Algorithm	0–20 s	20–40 s	40–60 s	60–80 s	80–100 s	Total Time (s)
Normal	77	155	215	272	323	1042
Quasi Steady	36	75	78	81	89	359
NOS	72	67	85	87	89	400

shows the time consumption of each period. The speed of the NOS algorithm is similar to that of the quasi-steady algorithm, which is 2.6 times that of the traditional coupling algorithm. In the period of 0–20 s, the calculation speed of the NOS is the same as that of the traditional algorithm, and the speed of the quasi-steady algorithm is twice that. In 20–100 s, the calculation speed of the optimization strategy is consistent with that of the quasi-steady algorithm, indicating that under the Nu change standard of 0.001, the coupling calculation time is shortened, and Nu tends to a stable state.

Figure 10 shows the Nu relative errors of the quasi-steady algorithm and the NOS algorithm after adding velocity perturbations in different time periods. The maximum relative error values of Nu for the three solids in the quasi-steady algorithm are 7.6%, 6.8%, and 6.0%, in that order, for the time period of 0–20 s. The time to reach the change standard specified by NOS is 11 s, and the maximum relative errors of Nu for the three solids are 1.3%, 1.1%, and 1.7%, respectively, which reduce the relative errors by 82.9%, 83.8%, and 71.7%, respectively, compared with the quasi-steady algorithm. In the time of 20–100 s, the calculation results of NOS and the quasi-steady algorithm are consistent, which indicates that the change standard of Nu does not capture the disturbance of the velocity boundary, and further indicates that the change in velocity has little impact on the heat transfer at the interface. This can also be confirmed by the change curve of average solid temperature.

Figure 11 shows the relative errors of the average solid surface temperature for the quasi-steady algorithm and the NOS algorithm after adding velocity perturbations in each time period. The relative errors of the three solid surface average temperatures during 0–20 s are −22.4%, −22.1%, and −22.0% for the quasi-steady algorithm and −0.6%, −1.3%, and −1.3% for the NOS algorithm, respectively. The NOS algorithm optimizes the relative errors of the quasi-steady algorithm by 97.3%, 94.1%, and 94.1%. Within 20–100 s, the absolute value of the error is stable within 2.5%, which means that the error will be controlled within a certain range when the speed boundary disturbance is added and the variation standard of Nu is 0.001.

5. Limitations

This study presents optimization for the initial error problem of the quasi-steady algorithm commonly used for conjugate heat transfer. The smaller the variation criterion used, the greater the optimization effect will be, but how to control the optimization effect within a certain range is still a problem that needs continued research.

6. Conclusions

The new optimization strategy of the quasi-steady algorithm proposed in this paper does not solve the steady-state flow field directly in the initial stage, but performs the coupling calculation. In the calculation, the strategy introduces a criterion for judging the stopping time of the coupled calculation for the first time, obtains the conjugate heat transfer characteristics, and determines the termination time of the flow field update according to the variation in the Nusselt number. In the validation case of unbounded disturbance, the relative error of the calculation results becomes smaller as the variation criterion is gradually reduced. The new optimization strategy reduces the relative error of the quasi-steady algorithm with respect to the mean temperature of the solid surface by 92.3% when a variation criterion of 0.001 is used. In the initial stage of calculation in the case with boundary perturbation, the relative errors of the three solid surface mean temperatures in the quasi-steady algorithm were reduced by 97.3%, 94.1%, and 94.1%, respectively, when the variation criterion of 0.001 was used for NOS, and then the absolute values of the relative errors were stabilized within 2.5%. The computational time consumed by the NOS and quasi-steady algorithms are similar, 400 s and 359 s, respectively, and the computational speed is 2.9 and 2.6 times that of the standard algorithm.

The variation criterion determined in the NOS substantially reduces the initial error of the quasi-steady algorithm while maintaining a similar computational speed to the quasi-steady algorithm, and the overall error can be controlled within a certain range.