Study on the sensitivity of steam ejector simulation to wall treatment methods

Considering the importance of turbulence model and wall treatment to steam ejector simulation, the influence of five wall treatment methods on numerical results was studied based on the realizable k-ε turbulence model (RKE). Combined with experimental data and SST turbulence model, the prediction of complex flow phenomena inside the ejector, including shock wave and reflux, by different wall treatment methods is discussed. The results show that under the numerical simulation conditions, the enhanced wall treatment method (EWT) combined with the RKE model can better predict the injection coefficient of the steam ejector, and the relative error compared with the experimental data is 5.3%. The standard wall function method (STWF) ignores the influence of the viscous bottom layer, resulting in inaccurate simulation of the internal flow phenomenon of the injector, and the injection coefficient calculated by the simulation is 10.2% higher than the experimental value. The results of the non-equilibrium wall function (NWF) method are relatively indifferent to the experimental values, and are not recommended when simulating steam ejectors.


Introduction
The steam ejector uses water vapor as the working fluid, which can use the residual pressure to recover low-pressure steam, relying on the emission of working steam and the mutual collision between fluids to transfer energy, without consuming external mechanical energy, and is a kind of fluid machinery that realizes mixing, heat exchange, pressurization and other functions [1].The steam ejector does not contain any components inside, which has the advantages of simple structure compared to other traditional fluid machinery.Steam ejectors are widely used in aerospace, chemical and refrigeration industries with their significant energy-saving effect, low environmental pollution working performance, low cost and convenient operation [2].
However, steam ejectors also have some defects, mainly manifested in low work efficiency, so many scholars have carried out research on steam ejectors.Entrainment rate (ER) and critical back pressure (CBP) are two important parameters used to describe steam ejector performance.Entrainment rate is defined as the ratio of the mass flow of the secondary stream to the mass flow of the primary stream, while the critical back pressure refers to the final pressure at which it will operate at its maximum capacity.For many years, computational fluid dynamics (CFD) techniques have been widely used in the performance prediction of steam ejectors and the analysis of internal flow fields [3].CFD has the advantages of low experimental cost, high safety performance, and accurate results, so many scholars use CFD to study steam ejectors [4], but many scholars have not reached a consensus on the selection of turbulence models, wall treatment and meshing when studying steam ejectors.
T. Sriveerakul [5] et al. used CFD to verify the performance prediction results of steam ejectors, and chose the realizable k-ε (RKE) turbulence model to simulate the flow phenomenon inside the ejector.This model is an improved version of the traditional "standard k-ε (SKE)" model, which is believed to be able to predict the diffusivity of circular jets more accurately than the traditional SKE model.The nonlinear control equation is solved "implicitly" by coupling, using the standard wall function method for wall treatment with y + of 30.Finally, the calculation error of the ER value is 12.69%, and the calculation error of the CBP value is 6.25%.
Amel Hemidi [6] et al. used CFD to verify the single-phase flow of steam ejectors when studying and analyzing steam ejector performance.The two turbulence models of k-ε and SST were used for comparison, and the extended wall function method (SCWF) was selected for the wall treatment of the k-ε turbulence model, and the y + was 11.The results show that the SST turbulence model simulates that the steam ejector has less dissipation, and the overall error of the k-ε model combined with the extended wall function method is less than 10%.
Satha Aphornratana [7] et al. used CFD to simulate the influence of the geometry of the main nozzle on jet refrigeration when studying the application of steam ejectors in the refrigeration cycle.The turbulent viscosity realizable k-ε (RKE) model was used for the calculation, a density-based solver was selected, and the standard wall function method (STWF) was used for the wall treatment with a y + of 30.Finally, compared with the experimental results, it is found that the calculation error of the simulated injection coefficient and back pressure is significantly different under different operating pressure conditions.Under the operating condition of 130° saturated steam in the primary flow, the relative errors of injection coefficient and critical outlet back pressure are 9.953% and 7.143%, respectively.
Yang [8] et al. analyzed the influence of different nozzle structures on the internal mixing process of the steam ejector, and selected three k-ε (SKE, RNG, RKE) turbulence models for the ejector simulation calculation, and the wall treatment all used the standard wall function method (STWF), and the y + was 20.It is found that the realizable k-ε (RKE) turbulence model has a better effect on the prediction of the injection coefficient and critical back pressure of the steam ejector.
Yinhai Zhu [5] et al. performed three-dimensional modeling of the ejector when studying the experimental and numerical simulation of the influence of shock wave characteristics on the performance of steam ejector.A pressure-based solver and an ideal gas model were selected to solve the nonlinear control equation, and the pressure field used the SIMPLEC algorithm.Three k-ε turbulence models and SST models were used for simulation calculation, and the standard wall function method (STWF) was selected for wall treatment when using the k-ε turbulence model for simulation calculation, with y + of 35.The results show that when the standard wall function method (STWF) is combined with the k-ε turbulence model, the prediction effect of shock wave is not as good as that of the SST model.
He Li [9] et al. studied the numerical simulation of the effect of superheating on the internal flow and entropy of the steam ejector of the MED-TVC desalination system, and the solver chose an implicit density solver based on the ideal gas model and the wet steam model.The SST model was chosen for the turbulence model, which did not require wall treatment.The results show that the SST model can better simulate the flow field inside the steam ejector, and the error of the simulation calculation is acceptable for both dry and wet steam.
When Yiqiao Li [10] et al. studied the double-resistance characteristics of the non-equilibrium condensing three-dimensional steam ejector, the finite volume method was used to solve the threedimensional steady-state N-S equation, and the convection term and diffusion term were used to use the second-order inverse style discretization, and the density solver was chosen to implicitly couple.The SST model is selected, and the results show that the SST model can better predict the blockage of the steam ejector in the diffusion section.
When Yuyan Hou [11] et al. studied the effect of main nozzle deviation on the performance of the steam ejector, they modeled the injector in three dimensions, selected the ideal saturated steam for the working medium, and selected the pressure-based solver to solve the discrete control equation.
Considering the compressible effect of gases, the turbulence model selection can be implemented k-ε (RKE) model, which uses the Standard Wall Function (STWF) method to deal with the interaction between steam and wall.The results show that the steam ejector with the main nozzle tilt is more likely to slide into subcritical mode, resulting in a decrease in the performance of the ejector.
Venkatesh Kakkirala [12] et al. proposed the optimal geometric parameters of steam ejectors when studying the optimal calculation method for the length and diameter of the mixing chamber of steam ejectors for gas turbine power plants.The ejector was modeled in 3D and a hexahedral structured mesh was created in ICEM CFD software.In the simulation calculation, the realizable k-ε (RKE) model was selected, the standard wall function method (STWF) was selected for wall treatment, y + was 30, and the pressure-velocity coupling was adopted by the SIMPLE algorithm.It was found that the length of the mixing chamber had little effect on the injector performance.
However, all this work did not reach consensus on the choice of numerical simulation methods for studying flow and heat transfer in steam ejectors, and furthermore, no special attention was paid to the influence of solvers or numerical simulation methods on the prediction results [8].Since the importance of turbulence model and wall treatment method and mesh to ejector simulation calculation cannot be ignored, and different wall treatment methods have different requirements for mesh, the sensitivity of steam ejector numerical simulation to wall treatment method is carried out.

Geometric models and experimental data
The typical steam ejector structure is mainly composed of four parts: main nozzle, mixing chamber, hose and subsonic diffuser, as shown in Figure 1(a) is a two-dimensional simplified model of the steam ejector.When the ejector is working, its process can be roughly divided into three stages: the first stage is that the high-pressure steam (primary fluid) passes through the ejector nozzle into supersonic steam, converting pressure energy into kinetic energy.The second stage is the mixing of working steam that becomes supersonic with the induced steam (secondary fluid), and the mass, energy and momentum exchange between the two, the kinetic energy of the induced steam is increased, and the working steam carries the induced steam into the diffusion stage.The third stage is the compression stage, in which the two vapors in the diffusion section continue to exchange energy, gradually compress kinetic energy and convert it into pressure energy, and finally discharge the mixed steam from the ejector outlet.In order to compare the simulation results of different wall treatment methods, the structure and operating conditions of steam ejectors studied by Xuelong Yang [8] et al. are selected as a benchmark.The structural parameters of the reference ejector are: nozzle throat diameter 2 mm, nozzle outlet diameter 8 mm, nozzle extension angle 10°, nozzle outlet position 35 mm, mixing chamber diameter 24 mm, hose diameter 19 mm, throat length 95 mm, diffusion section length 180 mm.Operating conditions: the primary flow is 130° steam at saturation temperature, the secondary flow is 10° steam at saturation temperature, and the back pressure is set to 3 kPa.

Standard Wall Function (STWF).
The wall function is based on the boundary layer theory, and the fluid has a boundary layer whether it is flow, heat transfer, and mass transfer.A characteristic of the boundary layer is that a certain physical quantity changes drastically, producing a very large gradient in the boundary layer, and the closer to the boundary layer, the greater the gradient, and outside the boundary layer, the physical quantity is almost the same as in the mainstream, and there is no gradient.In the midst of this problem, Fluent software provides a way to connect the viscous affected regions between walls and fully turbulent areas called wall functions, a semi-empirical formula.The standard wall function is the default wall function in Fluent software, which was proposed by Lauder and Spalding and is widely used for industrial fluid flow.The standard wall function is based on the assumption of constant shear stress and local equilibrium on the wall, and we try not to consider the use of the standard wall function in many calculations, most of which are used to compare the initial values of other wall treatments [13].
Standard wall functions use the typical logarithmic law: where k =0.4187 is the Ká rmá n constant; E =9.793 is the empirical constant; p U is the velocity immediately adjacent to the center of the wall mesh; p k is the turbulent kinetic energy immediately adjacent to the center of the grid;  is the dynamic viscosity.
U * and y * are dimensionless quantities in Fluent, equivalent to u + and y + .

Scalable Wall Function (SCWF).
Literally, the scalable wall function extends the standard wall function.When y + <11, the standard wall function cannot be used, and the scalable wall function can be used normally.The scalable wall function places certain restrictions on y + ; ) , (

Non-Equilibrium Wall Function (NWF).
Since the prediction results of both the standard wall function and the extended wall function are inaccurate when the pressure gradient of the wall is large, a new method is needed to solve this problem.
Fluent software provides a non-equilibrium wall function method that calculates wall shear stress   , turbulent kinetic energy k, and turbulent dissipation rate e based on two-layer assumptions.

Enhanced Wall Treatment (EWT).
The wall function only considers the scope of application of the logarithmic law, and completely ignores the influence of the sticky bottom.However, for some working cases, what we are concerned about is the change law of viscous underlying physical quantities, such as the phenomenon of boundary layer separation, when the wall function is no longer applicable.The method of enhanced wall treatment is to combine the two-layer model and the enhanced wall function, which does not produce large errors for the fully turbulent zone (y + >15) and the sticky bottom layer (y + ≈1) [14].

Menter-Lechner (M-L).
The method of using wall functions to treat walls with y + requirements is very demanding, and Menter-Lechner near-wall treatment is a model that is insensitive to y + .When the wall mesh is thin, a low Reynolds model is used.When the wall mesh is thick, use the wall function.The Menter-Lechner nearwall treatment method is designed to solve the problem of reinforced wall treatment at low Reynolds numbers.The Menter-Lechner near-wall treatment adds a source term to the transport equation for turbulent flow energy; In the formula,

Grid independence verification
Pianthong [15] et al. performed three-dimensional numerical calculations on supersonic steam ejectors and found no significant three-dimensional effects.When simulating numerical calculations for steam ejectors, the assumption of simplifying steam ejectors to axisymmetric is reasonable.Therefore, the computational mesh is created in a two-dimensional plane.The standard wall function requires that y + be greater than 30, and if it is lower than this value, the accuracy of the solution result will become very poor.Therefore, when drawing the grid, the wall should not be too dense, and the number of grids to ensure that y + is greater than 30 should not be controlled too much.Scalable wall functions and nonequilibrium wall functions do not require as much y + as standard wall functions.When y + is less than 11, the scalable wall function and the non-equilibrium wall function can be used normally, while the non-equilibrium wall function is more suitable when the wall pressure gradient is large.At this time, in order to capture more accurate wall flow information, the wall can be slightly encrypted.The two wall treatment methods of enhanced wall treatment and Menter-Lechner near-wall treatment do not have high requirements for y + , and Figure 2 shows the local magnification of the mesh with different wall treatment methods.In order to capture more accurate wall flow information, the wall mesh is encrypted.Different wall treatment methods have different requirements for meshes, and different wall function methods and reinforced wall treatments and Menter-Lechner near-wall treatments have different y + requirements.Therefore, in order to carry out the mesh-independent study, the mesh independence of different wall treatment methods was verified separately, and nine meshes were used for comparison, as shown in the following table [4]:  In numerical simulations, selecting a solver is the first step in starting the calculation, and many subsequent settings will correspond to the solver [16].In view of the pressure field extracted by manipulating the continuity equation in the pressure-based solver, a pressure-based solver was selected for the calculation in the simulation, and the pressure inlet and outlet were selected for both inlet and outlet [17].The iterative convergence criteria are set to have residual terms below 10 -6 and are stable, and the maximum axial velocity at the outlet of the steam ejector remains stable.Figure 3 shows the change of ER value with mesh number under a back pressure of 3 kPa.For different wall treatment methods, the mesh number of the final STWF consists of 78860 quadrilateral elements, the mesh number of SCWF and NWF consists of 110500 quadrilateral elements, and the mesh number of EWT, M-L and SST models consists of 192430 quadrilateral elements.

Calculation results and discussion
Figure 4 depicts the comparison of CFD calculations and experimental ER values for five wall treatment methods in the SST model and the realizable k-ε (RKE) model.It can be seen from the figure that when the back pressure of the steam ejector changes, the performance characteristics of the ejector are calculated by different turbulence models and wall treatment methods, and the critical back pressure (CBP) value of different wall treatment methods is different.Among the five wall treatment methods, the maximum ER value calculated by SCWF and NWF methods was the largest, and the relative errors compared with the experimental data were 32.1% and 24.6%, respectively.The maximum ER value calculated by the STWF method is larger than the experimental value, and the relative error is 12.6%, which further verifies that the STWF method should not be selected for wall treatment when simulating and calculating steam ejectors.The combination of EWT method and RKE model can better predict the performance of steam ejector, and the maximum ER value calculated by the M-L method simulation is higher than that of the EWT method, and the error compared with the experimental value is large, and it is not recommended.The SST model is more accurate and closer to the experimental values than the realizable k-ε (RKE) model for ejector simulations, which is consistent with the conclusions drawn in the literature [18].   2 lists the maximum Mach number numerical values calculated by the simulation of the five wall treatment methods in the SST model and the RKE model.Figure 5 shows that different wall treatments can simulate and calculate the first shock wave of the ejector at the nozzle outlet.The position of the first shock simulated by the five wall treatments remained approximately the same, except that the peak of the Mach number differed.The NWF and SCWF wall treatment methods predict the internal flow field of the injector is quite different from the other methods, and the intensity and peak prediction of the second shock wave are inaccurate, and it is not recommended to use it when simulating and calculating the injector.The SST model simulates that the first shock wave position is farther away from the nozzle outlet than the RKE model, and the shock wave moves towards the mixing chamber and the peak value is larger.Subsequently, the gas enters the waiting tube, and the second and third shocks appear, but the position of the shock simulated by different wall treatment methods and the peak of the Mach number are different.The Mach number simulated by the SST model did not change significantly in the section of the injector hose, while the other five wall treatment methods basically showed a sharp decline trend.Figure 6 shows that after the mixed gas enters the diffusion section, under the action of the pressure gradient, the SST model simulates the second shock sequence, at which time the pressure energy is converted to thermal energy, and the temperature of the mixed fluid rises rapidly [19].The EWT and STWF methods also showed shock waves in the diffusion section, and the error with the SST model was small, which further showed that the location and size of shock waves further affected the working performance of the steam ejector.), but the area of the vortex is smaller than that of the RKE model, and the energy of predicting the reflux loss is smaller, which further confirms that the prediction of the internal flow field is more accurate.The SCWF method simulates the reflux vortex in the mixing chamber (0.1 m~0.112 m), and the simulation calculation shows that the ER value is larger, the CBP value is small, and the result is quite different from the truth, so it is not recommended.The position of the reflux zone simulated by STWF and EWT method remained basically the same, and the reflux vortex occurred in the mixing chamber (0.069 m~0.132 m), and the EWT method also had a small reflux in the diffusion section (0.316 m~0.362 m), and the predicted internal flow field was closer to the real one, and the relative error between the ER value and CBP value and the experimental value was small.The NWF method simulates that the reflux vortex area occurs in the mixing chamber with the largest reflux vortex area, the largest reflux loss energy, and the smaller CBP value.The M-L method simulates the occurrence of reflux vortex in the diffusion section (0.301 m~0.328 m), and the area of the reflux vortex in the diffusion section is larger than that of the EWT method, the energy loss is larger, and the CBP value is smaller than the former, which further echoes the previous calculation results.The M-L method simulates the occurrence of reflux vortex in the diffusion section (0.301 m~0.328 m), the area of the reflux vortex in the diffusion section is larger than that of the EWT method, the energy loss is larger, and the CBP value is smaller than the former, which further echoes the previous calculation results.

Conclusion
Five wall treatment methods combined with the realizable k-ε (RKE) turbulence model and the SST model were combined to obtain a combination of six simulation methods to simulate the performance and internal flow phenomenon of the steam ejector, and the calculation results were compared with the experimental data.The following conclusions are drawn: (1) The STWF method combined with the RKE model to simulate the performance of the steam ejector and the internal flow field error is larger than that of the SST model, and the relative error between the ER value and the experimental value calculated by the simulation is 12.6%, and the CBP value is slightly smaller than the experimental value.The STWF method is based on the assumption of constant shear force and local equilibrium on the wall, ignoring the complex flow of the viscous bottom layer (y + ≈1).The internal wall flow process of the steam ejector is very complicated, and there are complex mixed shear layers, and the STWF method has very high requirements for y+, and it is necessary to strictly control y + to calculate it under the condition of greater than 30.Therefore, it is not recommended to use the standard wall function (STWF) method to simulate the internal flow phenomenon of the injector, the simulation of the calculated ejector performance is not accurate enough, and the STWF method is often used to compare the initial value with other methods.(2) When simulating steam ejector performance and internal flow field, if you consider not using wall treatment, it is recommended to directly select the SST model, and the results of the SST model simulation are closer to the experimental values.If wall treatment is considered, among the five wall treatment methods, the enhanced wall treatment method (EWT) combined with the RKE model combined with the simulation results are more accurate.Therefore, when simulating the steam ejector performance and internal flow for the turbulence model using the e equation, it is recommended to directly use the enhanced wall treatment to try not to use the wall function, the enhanced wall treatment method has higher requirements for the mesh than the wall function, and the wall can be encrypted when drawing the grid to obtain more accurate internal flow field information.
(3) Compared with the other methods, the SCWF and NWF methods had the largest relative errors between the ejector performance and experimental values, which were 32.1% and 24.6%, respectively.These two methods are slightly extended on the basis of STWF, and the NWF method is based on the assumption of two layers of the wall to simulate the calculation, which is very different from the real value.Therefore, these two wall treatments are not recommended when using the RKE model to simulate the calculation of steam ejectors.

Figure 1 (Figure 1 .
Figure 1.(a)Two-dimensional simplified model of steam ejector(b) Illustrative performance curve of the steam ejector.An important parameter used to describe the performance of an ejector is"an entrainment ratio"[5]:

S
exists only in the viscous low layer and is used instead of the low Reynolds number model, while in the logarithmic region, wall near − S automatically becomes zero.

Figure 2 .
Figure 2. Local meshes for different wall treatments.

Figure 3 .
Figure 3.The ER value varies with the number of grids under the condition of back pressure of 3 kPa.

Figure 4 .
Figure 4. Comparison of injector performance with CFD results of different wall treatments and experimental results.

Figures 5
Figures 5 and 6 depict the five wall treatment methods of the realizable k-ε (RKE) model under a back pressure of 3 kPa, respectively, and the Mach number change curve and the Mach number cloud of the internal flow field along the central axis of the steam ejector.Table 2 lists the maximum Mach number numerical values calculated by the simulation of the five wall treatment methods in the SST model and the RKE model.Figure5shows that different wall treatments can simulate and calculate the first shock wave of the ejector at the nozzle outlet.The position of the first shock simulated by the five wall treatments remained approximately the same, except that the peak of the Mach number differed.The NWF and SCWF wall treatment methods predict the internal flow field of the injector is quite different from the other methods, and the intensity and peak prediction of the second shock wave are inaccurate, and it is not recommended to use it when simulating and calculating the injector.The SST model simulates that the first shock wave position is farther away from the nozzle outlet than the RKE model, and the shock wave moves towards the mixing chamber and the peak value is larger.Subsequently, the gas enters the waiting tube, and the second and third shocks appear, but the position of the shock simulated by different wall treatment methods and the peak of the Mach number are different.The Mach number simulated by the SST model did not change significantly in the section of the injector hose, while the other five wall treatment methods basically showed a sharp decline trend.Figure6shows that after the mixed gas enters the diffusion section, under the action of the pressure gradient, the SST model simulates the second shock sequence, at which time the pressure energy is converted to thermal energy, and the temperature of the mixed fluid rises rapidly[19].The EWT and STWF methods also showed shock waves in the diffusion section, and the error with the SST model was small, which further showed that the location and size of shock waves further affected the working performance of the steam ejector.Figure 7 depicts the wall pressure distribution and experimental data of the five wall treatment methods of SST model and RKE model under the condition of back pressure of 3 kPa.It can be seen from the figure that the pressure distribution curves simulated by the EWT and STWF methods are more consistent with the experimental value distribution, and the simulation results of SCWF and NWF methods are quite different from the experimental values, which echoes the previous calculation results [20].

Figure 7
Figures 5 and 6 depict the five wall treatment methods of the realizable k-ε (RKE) model under a back pressure of 3 kPa, respectively, and the Mach number change curve and the Mach number cloud of the internal flow field along the central axis of the steam ejector.Table 2 lists the maximum Mach number numerical values calculated by the simulation of the five wall treatment methods in the SST model and the RKE model.Figure5shows that different wall treatments can simulate and calculate the first shock wave of the ejector at the nozzle outlet.The position of the first shock simulated by the five wall treatments remained approximately the same, except that the peak of the Mach number differed.The NWF and SCWF wall treatment methods predict the internal flow field of the injector is quite different from the other methods, and the intensity and peak prediction of the second shock wave are inaccurate, and it is not recommended to use it when simulating and calculating the injector.The SST model simulates that the first shock wave position is farther away from the nozzle outlet than the RKE model, and the shock wave moves towards the mixing chamber and the peak value is larger.Subsequently, the gas enters the waiting tube, and the second and third shocks appear, but the position of the shock simulated by different wall treatment methods and the peak of the Mach number are different.The Mach number simulated by the SST model did not change significantly in the section of the injector hose, while the other five wall treatment methods basically showed a sharp decline trend.Figure6shows that after the mixed gas enters the diffusion section, under the action of the pressure gradient, the SST model simulates the second shock sequence, at which time the pressure energy is converted to thermal energy, and the temperature of the mixed fluid rises rapidly[19].The EWT and STWF methods also showed shock waves in the diffusion section, and the error with the SST model was small, which further showed that the location and size of shock waves further affected the working performance of the steam ejector.Figure 7 depicts the wall pressure distribution and experimental data of the five wall treatment methods of SST model and RKE model under the condition of back pressure of 3 kPa.It can be seen from the figure that the pressure distribution curves simulated by the EWT and STWF methods are more consistent with the experimental value distribution, and the simulation results of SCWF and NWF methods are quite different from the experimental values, which echoes the previous calculation results [20].

Figure 5 .
Figure 5.The CFD results of different wall treatment methods were distributed along the central axis of the ejector under the condition of back pressure of 3 kPa.

Figure 6 .
Figure 6.Mach number distribution cloud under back pressure of 3 kPa under different wall treatment methods(a-f).

Figure 7 .
Figure 7.The wall pressure distribution of different wall treatment methods under the condition of back pressure of 3 kPa was compared with experimental data.

Figure 8
Figure 8 depicts the comparison of the reflux zone of the internal flow field of the steam ejector under the condition of 3 kPa under the condition of 3 kPa backpressure for five wall treatment methods of SST model and RKE model.It can be seen from the figure that the SST model and the RKE model simulate the reflux of the internal flow field of the injector with obvious differences, and the SST model has two reflux vortices in the diffusion section (0.319 m~0.412 m), but the area of the vortex is smaller than that of the RKE model, and the energy of predicting the reflux loss is smaller, which further confirms that the prediction of the internal flow field is more accurate.The SCWF method simulates the reflux vortex in the mixing chamber (0.1 m~0.112 m), and the simulation calculation shows that the ER value is larger, the CBP value is small, and the result is quite different from the truth, so it is not recommended.The position of the reflux zone simulated by STWF and EWT method remained basically the same, and the reflux vortex occurred in the mixing chamber (0.069 m~0.132 m), and the EWT method also had a small reflux in the diffusion section (0.316 m~0.362 m), and the predicted internal flow field was closer to the real one, and the relative error between the ER value and CBP value and the experimental value was small.The NWF method simulates that the reflux vortex area occurs in the mixing chamber with the largest reflux vortex area, the largest reflux loss energy, and the smaller CBP value.The M-L method simulates the occurrence of reflux vortex in the diffusion section (0.301 m~0.328 m), and the area of the reflux vortex in the diffusion section is larger than that of the EWT method, the energy loss is larger, and the CBP value is smaller than the former, which further echoes the previous calculation results.The M-L method simulates the occurrence of reflux vortex in the diffusion section (0.301 m~0.328 m), the area of the reflux vortex in the diffusion section is larger than that of the EWT method, the energy loss is larger, and the CBP value is smaller than the former, which further echoes the previous calculation results.

Figure 8 .
Figure 8.Comparison of reflux zones under the condition of 3 kPa back pressure of different wall treatment methods (a-f).

Table 1 .
Mesh independence validation table.

Table 2 .
Maximum Mach number under the condition of 3 kPa under back pressure for different wall treatment method.