High-precision prediction method for high-speed turbulent boundary layer friction based on compressible transformation

Abstract

This invention discloses a high-precision prediction method for high-speed turbulent boundary layer friction based on compressible transformation, belonging to the field of fluid mechanics engineering calculation and experimental technology. Specifically, the method involves: 1) Obtaining the basic parameters of the zero-pressure gradient compressible flat plate turbulent boundary layer to be predicted, including the incoming Mach number, momentum-thickness Reynolds number, Prandtl number, incoming reference temperature, wall temperature, recovery temperature, and viscosity index; 2) Setting the initial iteration values, convergence criteria, and maximum iteration steps for numerical calculation; 3) Pre-calculating the values ​​of the auxiliary function in the improved drag transformation model over the specified interval, as well as the ratio of free-flow velocities before and after the transformation; 4) After the pre-calculation, using an iterative method to solve for the target parameters in the improved drag transformation model; 5) Calculating the wall friction coefficient and heat flux coefficient based on the obtained target parameters and basic parameters. This method can achieve rapid and high-precision prediction of the wall friction coefficient and heat flux coefficient under a wide range of Mach numbers and complex wall heat transfer conditions, demonstrating significant engineering practicality.

Description

Technical Field

[0001] This invention belongs to the field of fluid mechanics engineering calculation and experimental technology, specifically relating to a high-precision prediction method for high-speed turbulent boundary layer friction based on compressible transformation. Background Technology

[0002] Compressible wall turbulence has significant applications in many engineering fields such as aerospace and energy, including aerodynamic heating analysis of aircraft surfaces, hypersonic inlet design, and thermal protection system evaluation. Accurate prediction of wall friction and heat flux in zero-pressure-gradient compressible flat plate boundary layer turbulence is crucial for optimizing engineering design and improving system performance and safety. Currently, the prediction of wall friction in zero-pressure-gradient compressible flat plate boundary layer turbulence mainly relies on three technical approaches: traditional numerical simulation, experimental measurement, and the classic van Driest drag transformation method. However, each approach has its limitations.

[0003] Traditional numerical simulation methods include Direct Numerical Simulation (DNS) and Reynolds-Averaged Navier-Stokes (RANS) models. DNS, based on the complete Navier-Stokes equations, accurately captures the anisotropic turbulent characteristics of the flow field across all scales, providing highly detailed and accurate flow field information. However, this method is limited by enormous computational resource requirements; the number of computational grids increases exponentially with the Reynolds number. In three-dimensional high Reynolds number boundary layer simulations at the engineering scale, the computational cost is prohibitively high, completely failing to meet the rapid iteration requirements of practical engineering designs. While the Reynolds-Averaged Navier-Stokes model reduces computational load to some extent, due to the inherent limitations of the closed-loop theory of turbulence models, its calculation results have relatively large errors, especially under high Mach numbers and strong wall cooling conditions, making it difficult to meet the requirements for high-precision predictions.

[0004] Existing experimental measurement techniques such as hot wire anemometers (HWA) and particle image velocimetry (PIV) face several bottlenecks in compressible flows: (1) High-speed flow causes high-frequency response lag in sensors or poor tracking of tracer particles, making it impossible to obtain flow field information in a timely and accurate manner; (2) The cost of a single experiment is extremely high, and the effective operating time in a high-enthalpy wind tunnel is extremely short, which greatly limits the range and repeatability of experimental data; (3) Flow measurement very close to the wall is often very difficult, hot wire probes will interfere with the flow field, and particle image velocimetry will introduce measurement distortion due to the sparse tracer particles in the near-wall region.

[0005] The classic van Driest drag transformation, based on the van Driest velocity transformation method, aims to map a compressible zero-pressure gradient flat plate turbulent boundary layer to its corresponding incompressible state. However, in practical engineering applications, it has been found that the classic van Driest II transformation and its subsequent derived empirical correction models (such as the Spalding-Chi transformation) exhibit high data dispersion in posterior predictions, especially under high Mach number and low wall temperature conditions.

[0006] Therefore, there is an urgent need to develop a method that can rapidly, stably, and accurately predict the wall friction and heat flux coefficient of a compressible turbulent boundary layer with zero pressure gradient, based solely on input basic flow parameters (including incoming Mach number, Reynolds number based on incoming conditions and momentum thickness, Prandtl number, incoming reference temperature, wall temperature, recovery temperature, and the exponent of the exponential viscosity model) over a very wide range of Mach numbers and under complex wall heat transfer conditions. Summary of the Invention

[0007] The purpose of this invention is to overcome the problems of high computational cost and insufficient accuracy in the prediction of wall friction and heat flux coefficient in compressible turbulent boundary layer with zero pressure gradient by traditional methods, and to provide a high-precision prediction method for wall friction and heat flux coefficient in high-speed flat plate turbulent boundary layer based on compressible transformation, so as to realize the rapid and accurate prediction of wall aerodynamic force and aerothermal force under complex working conditions with only basic flow parameters.

[0008] The specific technical solution adopted in this invention is as follows:

[0009] In a first aspect, the present invention provides a high-precision prediction method for high-speed turbulent boundary layer friction based on compressible transformation, the specific steps of which are as follows:

[0010] S1: Obtain the basic parameters of the zero-pressure gradient compressible flat plate boundary layer turbulence to be predicted; the basic parameters include the incoming Mach number, momentum-thickness Reynolds number, Prandtl number, incoming reference temperature, wall temperature, recovery temperature, and viscosity index;

[0011] S2: Set the initial iteration value, convergence criterion, and maximum iteration step for numerical computation;

[0012] S3: First auxiliary function in the pre-calculated improved drag transformation model Second auxiliary function exist The values ​​in the interval, and the ratio F of the free flow velocities before and after the transformation;

[0013] S4: After the pre-calculation is completed, the target parameters in the improved drag transformation model are solved by iterative method;

[0014] S5: Based on the obtained target parameters and basic parameters, calculate the wall friction coefficient and heat flux coefficient.

[0015] As a preferred embodiment, the improved drag transformation model described in step S3 is as follows:

[0016] ;

[0017] ;

[0018]

[0019] ;

[0020] In the formula: This represents the coefficient of incompressible wall friction corresponding to the transformation. Let be the wall friction coefficient to be solved; This is the friction coefficient transformation factor; The Reynolds number representing the incompressible momentum thickness corresponding to the transformation. The compressible momentum thickness Reynolds number; The momentum-thickness Reynolds number transformation factor; For free flow density; The wall density; This represents the ratio of the free-flow velocities before and after the transformation. The viscosity coefficient of the incoming flow; The wall viscosity coefficient; The velocity ratio at the edge of the compressible boundary layer; is the equivalent incompressible boundary layer edge velocity ratio; 'a' is the objective parameter to be solved. and These represent the first and second auxiliary functions, respectively; n is the exponent of the viscous power-law model; B and -A 2 These are the coefficients of the first and second terms in the average temperature-velocity relationship, respectively.

[0021] Furthermore, the pre-calculation described in step S3 is as follows:

[0022] First, calculate parameters B and A. 2 :

[0023] ;

[0024] ;

[0025] In the formula: s is the Reynolds analogy factor, with a value of 1.14; Pr is the Prandtl number; T r To restore temperature; This refers to the wall temperature at the corresponding flow direction location; Reference temperature for incoming flow;

[0026] Then the calculated B and A 2 And the exponent n of the viscous power-law model is substituted into the first auxiliary function. The second auxiliary function is obtained by numerical integration or by analytical solution of the incomplete β function. Finally, the ratio of the free-flow velocity before and after the transformation is calculated. .

[0027] Furthermore, in step S4, Newton's iteration method is used for solving the problem. The specific steps are as follows:

[0028] S41: Based on the current iteration value Calculate the integral variable;

[0029] ;

[0030] ;

[0031] In the formula: and , respectively, are the first intermediate integral and the second intermediate integral; z is the dimensionless average velocity of the free flow velocity in the compressible case, and Z is the dimensionless average velocity of the free flow velocity in the equivalent incompressible case;

[0032] S42: Calculate the Reynolds number for the current step's incompressible momentum thickness. The calculation formula is as follows:

[0033] ;

[0034] S43: Calculate the target value a target The calculation formula is as follows:

[0035]

[0036] In the formula: The equivalent logarithmic law slope constant is 0.375. is the Kármán constant in the Coles-Fernholz formula, with a value of 0.384; This is the intercept constant in the Coles-Fernholz formula, with a value of 4.127;

[0037] S44: Calculate the current iteration value and target value a target deviation If the deviation is less than the convergence criterion, then it is considered convergent, and the current iteration value is set. The objective parameter 'a' is used as the solution; otherwise, the deviation is calculated. about The derivative of the value is calculated, and the iteration value is updated using Newton's iteration method; steps S41 to S43 are repeated until convergence or the maximum number of iterations is reached.

[0038] Furthermore, the wall friction coefficient C mentioned in step S5 f The calculation formula is as follows:

[0039] ;

[0040] In the formula: a is the target parameter obtained by solving. The equivalent logarithmic law slope constant, For the incoming flow reference temperature, This refers to the wall temperature.

[0041] Furthermore, the heat flux coefficient B mentioned in step S5 q The calculation formula is as follows:

[0042] ;

[0043] In the formula: C f T is the wall friction coefficient; r To restore temperature, For the incoming flow reference temperature, Let be the wall temperature, and s be the Reynolds analogue factor.

[0044] Preferably, the initial iteration value in step S2 is set to 10, and the convergence criterion is set to 10. -8 The maximum number of iterations is set to 50.

[0045] Secondly, the present invention protects an application of the prediction method described in the first aspect, which predicts the wall friction coefficient and heat flux coefficient, and then uses them for a benchmark assessment of the aerodynamic friction and heat flux on the surface of a high-speed aircraft to guide the preliminary design of the aerodynamic shape and thermal protection system.

[0046] Compared with the prior art, the present invention has the following advantages:

[0047] The prediction method provided by this invention can quickly obtain the wall friction coefficient and heat flux coefficient based on the basic flow parameters of the turbulent boundary layer of a compressible flat plate with zero pressure gradient under complex working conditions, and has significant efficiency and accuracy.

[0048] The prediction method provided by this invention significantly improves the prediction accuracy of flows under high Mach number and cold wall temperature conditions, offering timely data support for engineers and researchers. It enables a comprehensive and accurate understanding of wall friction and thermal flux states in complex flows within a short time, significantly enhancing the reliability and efficiency of complex flow analysis. This rapid response capability not only improves work efficiency but also provides strong support for timely design optimization and parameter adjustment, avoiding the cumbersome process of obtaining complete flow field data through multiple simulations required by traditional methods. Therefore, it provides a more convenient, efficient, and reliable means for the research and application of zero-pressure gradient compressible flat plate boundary layer flows. Attached Figure Description

[0049] Figure 1 A flowchart of the prediction method provided by the present invention;

[0050] Figure 2 This is a graph showing the relationship between the incompressible wall friction coefficient and the corresponding incompressible momentum-thickness Reynolds number calculated under a wide range of Mach numbers, wall temperature parameters, and momentum-thickness Reynolds numbers in Example 1.

[0051] Figure 3 This is a graph showing the relative percentage error between the predicted compressible wall friction coefficients in Example 1 under a wide range of Mach numbers, wall temperature parameters, and momentum-thickness Reynolds numbers and the direct numerical simulation (DNS) reference results. Detailed Implementation

[0052] The present invention will be further described and illustrated below with reference to the accompanying drawings and specific embodiments. The technical features of each embodiment of the present invention can be combined accordingly, provided that there is no mutual conflict.

[0053] like Figure 1 As shown in the figure, as one specific embodiment of the present invention, a high-precision prediction method for high-speed turbulent boundary layer friction based on compressible transformation is provided, as follows:

[0054] Step 1: Obtaining Basic Parameters

[0055] The basic parameters of the zero-pressure gradient compressible flat plate boundary layer turbulence to be predicted are obtained. In this embodiment, the basic parameters include the incoming Mach number. Compressible momentum, thickness, Reynolds number Prandtl number Pr, incoming flow reference temperature Wall temperature Recovery temperature T r And the viscosity exponent n in the power-law viscosity model.

[0056] ①The power-law viscosity model is as follows:

[0057] ;

[0058] In the formula: and These represent the local average kinetic viscosity and average temperature, respectively. and These represent the corresponding wall dynamic viscosity and wall temperature, respectively; n is the viscosity index.

[0059] The power-law viscosity model describes the relationship between viscosity coefficient and temperature. The value of the viscosity index is usually determined according to the type of gas, and for air, it is usually taken as 0.75.

[0060] ②Recovery temperature T r It can be calculated using the following formula:

[0061] ;

[0062] In the formula: M is the reference temperature for the incoming flow. ∞ The Mach number of the incoming flow; For specific heat ratio, for air ; r is the recovery factor, for turbulent boundary layers, .

[0063] Step 2: Construct an improved resistance transformation model;

[0064] (1) Assume the wall friction coefficient for compressible flow With momentum, thickness, and Reynolds number It can be achieved through two transformation factors and Mapped to the equivalent incompressible wall friction coefficient and equivalent incompressible momentum thickness Reynolds number :

[0065] ;

[0066] In the formula: Represents the equivalent incompressible wall friction coefficient. Let be the coefficient of friction of the compressible wall to be solved; This is the friction coefficient transformation factor; Represents the equivalent incompressible momentum thickness Reynolds number. The input is the compressible momentum thickness Reynolds number; is the momentum-thickness Reynolds number transformation factor.

[0067] The purpose of this mapping is to transform the complex problem of predicting compressible flows into solving the scaling law problem of mature incompressible flows.

[0068] (2) For an equivalent zero-pressure gradient incompressible flat plate turbulent boundary layer, there is a mature scaling law relationship between wall friction and momentum-thickness Reynolds number, namely the Coles-Fernholz relation:

[0069] ;

[0070] In the formula: , These are the empirical parameters used for calibration.

[0071] (3) Determine the compressible transformation factor and The exact expression:

[0072] ;

[0073] In the formula: For free flow density; The wall density; For free-flow dynamic viscosity; Let be the wall dynamic viscosity; F be the ratio of free flow velocities; and I and J be the first and second intermediate integrals, respectively, containing the objective parameter a to be solved.

[0074] (4) Using the generalized Reynolds analogy theory, the average temperature within the boundary layer is established. The analytical relationship between the dimensionless average velocity z of free flow in the compressible case and the following:

[0075] ;

[0076] ;

[0077] In the formula: For free flow velocity; The local flow-averaged velocity within the boundary layer; B and -A 2 These are the coefficients of the linear term and the intermediate coefficients of the quadratic term, respectively.

[0078] Coefficient A 2 B is calculated from the obtained basic parameters using the following formula:

[0079] ;

[0080] ;

[0081] In the formula: s is the Reynolds analogue factor, usually taken as 1.14; Pr is the Prandtl number; T r To restore temperature; The wall temperature; The reference temperature for the incoming flow.

[0082] (5) Using the above-mentioned average temperature-velocity analytical relationship and power-law viscosity model, construct the first auxiliary function. Second auxiliary function :

[0083] ;

[0084] ;

[0085] In the formula: n is the exponent in the power-law viscosity model.

[0086] At the same time, F is defined as a function derived from the second auxiliary function. A definite constant, i.e., the ratio of free flow velocities .

[0087] (6) Define the momentum-thickness Reynolds number transformation factor The integrals appearing in the text are the first intermediate integral I and the second intermediate integral J, both of which take the target parameter a as the variable:

[0088] ;

[0089] ;

[0090] In the formula: This represents the dimensionless average velocity at the edge of the compressible boundary layer. , which is the equivalent incompressible boundary layer edge velocity ratio, and serves as the upper limit of the integral I.

[0091] (7) Finally, the improved drag transformation model constructed in this embodiment is as follows:

[0092] ;

[0093] ;

[0094] ;

[0095] ;

[0096] In actual solution, the value of the objective parameter 'a' is adjusted using Newton's iteration method to make the transformed equivalent parameter... and Strictly satisfies the scaling law of incompressible boundary layers.

[0097] Step 3: Initialize the iterative solver and set the initial iteration value 'a' for numerical computation. (0) =10, convergence tolerance ε=10 -8 and maximum number of iterations N max =50.

[0098] Step 4: Pre-calculate the first auxiliary function in the improved drag transformation model. Second auxiliary function exist Values ​​in the interval, and free flow velocity ratio .

[0099] In specific implementation, (in Divide the interval into N equal subintervals (e.g., N=2000), and use the composite trapezoidal rule to... Numerical integration is performed to obtain the function values ​​at each discrete point; simultaneously, calculations are performed. . Similarly, calculate the function value at the same discrete point and store it for use in subsequent steps.

[0100] Step 5: Solve the improved drag transformation model using Newton's iteration method to determine the objective parameter 'a', as follows:

[0101] (1) Based on the parameter values ​​of the current iteration step Calculate the following integrals and their relationship with respect to parameters. The derivative:

[0102] ;

[0103] ;

[0104] in, The dimensionless average velocity at the edge of the compressible boundary layer. The integral is obtained from the calculation results in the above steps. The above integral can be calculated using numerical integration methods, specifically the second intermediate integral. It can also be obtained analytically using integration by parts.

[0105] According to the definition, the corresponding derivative and It can be calculated using the following formula:

[0106] ;

[0107] ;

[0108] Similarly, the above derivatives can be calculated using numerical integration methods, and It can also be obtained through analysis.

[0109] (2) Based on the current step and Calculate the Reynolds number of the equivalent incompressible momentum thickness corresponding to the current step. The calculation formula is as follows:

[0110] ;

[0111] In the formula: For the dynamic viscosity of free flow The ratio of the kinetic viscosity at the wall to the viscosity of the other two is given. This can be achieved through the aforementioned power-law viscosity model and temperature ratio. calculate.

[0112] (3) Based on the incompressible Coles-Fernholz scaling law, calculate the target value a of the iterative parameters. target The calculation formula is as follows:

[0113]

[0114] In the formula: The equivalent logarithmic law slope constant is 0.375. is the KAMAN constant for the incompressible Coles-Fernholz scaling law, with a value of 0.384; is the intercept constant of the incompressible Coles-Fernholz scaling law, with a value of 4.127;

[0115] Calculate the current iteration value and target value a target deviation :

[0116] .

[0117] (4) Convergence judgment and update

[0118] Determine if the residuals meet the convergence condition: If If the iteration is successful, the current iteration value is determined to be converged. The final target parameter 'a' is used as the objective parameter; otherwise, the residuals are calculated with respect to the parameter. derivative :

[0119] ;

[0120] The next iteration value is updated using Newton's iteration method with a relaxation factor. :

[0121] ;

[0122] In the formula, is the relaxation factor, with a value ranging from 0.5 to 1.0. In this embodiment, it is taken as... .

[0123] Repeat steps (1) to (4) above until the convergence condition is met or the maximum number of iterations N is reached. max =50.

[0124] Step 6: Based on the target parameter 'a' and basic parameters obtained through iterative solutions, calculate the wall friction coefficient. and heat flux coefficient The calculation formulas are as follows:

[0125] ;

[0126] In the formula: a is the target parameter obtained by solving. The equivalent logarithmic law slope constant, For the incoming flow reference temperature, This refers to the wall temperature.

[0127] ;

[0128] In the formula: C f The wall friction coefficient is obtained from the preceding formula; T r To restore temperature; For the incoming flow reference temperature, is the wall temperature; s is the Reynolds analogue factor, which is usually taken as 1.14.

[0129] Thus, the wall friction coefficient of a compressible flat plate boundary layer with zero pressure gradient was determined. and heat flux coefficient High-precision prediction.

[0130] Example 1

[0131] To verify the prediction accuracy of the prediction method provided by this invention, this embodiment selects different operating conditions with different incoming Mach numbers and wall temperatures for testing.

[0132] (1) Operating conditions

[0133] ①Incoming Mach number In the formula:

[0134] In this embodiment, the Mach number of the incoming flow The value ranges from 0.3 to 13.64, covering subsonic, transonic, supersonic, and hypersonic speeds.

[0135] ② Wall temperature parameters In the formula: The reference temperature for free flow. The wall temperature, To restore temperature.

[0136] Wall temperature parameters in this embodiment The value ranges from -0.55 to 2.85, covering extreme cold wall conditions. ) and hot walls ( )condition.

[0137] (2) Verification of transformation relationship

[0138] To verify the accuracy and physical consistency of the improved drag transformation model constructed in this invention, a high-precision direct numerical simulation (DNS) database was used to perform a forward verification of the mapping relationship. The specific verification steps are as follows:

[0139] ①DNS data extraction and preparation:

[0140] Benchmark data for different operating conditions were extracted from a peer-reviewed, high-precision compressible zero-pressure gradient flat plate turbulent boundary layer DNS database. The extracted data includes not only the aforementioned basic flow parameters (incoming Mach number)... Compressible momentum, thickness, Reynolds number Prandtl number Pr, incoming flow reference temperature Wall temperature Recovery temperature T r In addition to the viscosity index n), it also includes the actual compressible wall friction coefficient C. f .

[0141] ② Calculate the forward transformation factor:

[0142] Using the above DNS real C f Data, used to deduce the precise value of the target parameter 'a' (which can be directly derived from the formula). The calculation is analytical, eliminating the need for the Newton iterations described in the previous embodiments. Substituting this parameter 'a' and the basic flow parameters into the auxiliary function defined in this invention... and And in the intermediate integral variables I and J, the corresponding friction coefficient transformation factor is calculated. Momentum-thickness Reynolds number transformation factor .

[0143] ③ Perform forward physics mapping:

[0144] Using the calculated transformation factor, the compressible wall friction in the DNS database is... and momentum thickness Reynolds number The following formula provides a positive mapping to the equivalent incompressible space:

[0145] ;

[0146] This allows us to obtain the equivalent incompressible wall friction coefficients for various complex working conditions. and equivalent incompressible momentum thickness Reynolds number .

[0147] ④ Compare with the standard scaling law for judgment:

[0148] The verification results are as follows Figure 2 As shown, the compressible data points obtained after transformation are highly folded and conform to the theoretical scaling law of incompressible zero-pressure gradient turbulent boundary layers, namely the classic Coles-Fernholz relation (black dashed line):

[0149] ;

[0150] in, , The results show that, under different Mach numbers, wall temperatures, and Reynolds numbers, the data after forward transformation all exhibit extremely high physical consistency with classical theory, and the data dispersion is extremely low.

[0151] (3) Validation of prediction accuracy

[0152] Based on the strict validity of the above transformation relationship, the method of the present invention is further verified to affect the coefficient of friction of compressible walls. The accuracy of direct predictions. Using the same batch of direct numerical simulation (DNS) data as a reference, the accuracy of direct predictions obtained through iterative solutions in this embodiment of the invention is compared. Compare with the actual DNS results and calculate the relative percentage error.

[0153] The verification results are as follows Figure 3 As shown, for the vast majority of operating conditions, the relative prediction error of the method of this invention is strictly controlled within ±5%. Especially under extreme conditions such as high Mach numbers and strong cold wall conditions, where traditional empirical transformation methods are prone to large deviations, the method of this invention still maintains extremely high robustness and accuracy. Statistical analysis shows that the average prediction error for all covered operating conditions is only 3.08%, and its overall prediction accuracy is significantly better than classic drag transformation methods such as van Driest-II.

[0154] In summary, the proposed method for predicting wall friction in zero-pressure gradient compressible flat plate boundary layer turbulence based on VIPL transform can achieve high-precision, low-dispersion prediction of the wall friction coefficient over a wide range of incoming Mach number, wall temperature, and Reynolds number. This method is based on rigorous theory and incorporates a Newton-Raphson iterative solution strategy, resulting in high computational efficiency. It is suitable for rapid aerodynamic and thermodynamic evaluation applications in modern aerospace engineering and lays a reliable physical and mathematical foundation for the accurate prediction of subsequent wall heat flux coefficients.

[0155] The embodiments described above are merely preferred embodiments of the present invention and are not intended to limit the invention. Those skilled in the art can make various changes and modifications without departing from the spirit and scope of the invention. Therefore, all technical solutions obtained through equivalent substitution or transformation fall within the protection scope of the present invention.

Claims

1. A high-precision prediction method for high-speed turbulent boundary layer friction based on compressible transformation, characterized in that, The specific steps are as follows: S1: Obtain the basic parameters of the zero-pressure gradient compressible flat plate turbulent boundary layer to be predicted; the basic parameters include the incoming Mach number, momentum-thickness Reynolds number, Prandtl number, incoming reference temperature, wall temperature, recovery temperature, and viscosity index; S2: Set the initial iteration value, convergence criterion, and maximum iteration step for numerical computation; S3: Pre-computing first auxiliary function in improved drag transformation model and second auxiliary function In interval, and the ratio of free stream velocities before and after transformation F; S4: After the pre-calculation is completed, the target parameters in the improved drag transformation model are solved by iterative method; S5: Based on the obtained target parameters and basic parameters, calculate the wall friction coefficient and heat flux coefficient.

2. The high-accuracy prediction method of compressible -transform-based high-speed turbulent boundary layer friction drag according to claim 1, characterized in that, The improved drag transformation model described in step S3 is as follows: ； ； 3. ； wherein: represents the transformed non-compressible wall skin friction coefficient corresponding to the transformation, is the wall skin friction coefficient to be solved; is the skin friction coefficient transformation factor; represents the transformed non-compressible momentum thickness Reynolds number corresponding to the transformation, is the compressible momentum thickness Reynolds number; is the momentum thickness Reynolds number transformation factor; is the free stream density; is the wall density; is the ratio of free stream velocities before and after the transformation; is the free stream viscosity coefficient; is the wall viscosity coefficient; is the velocity ratio of the compressible boundary layer edge; is the equivalent non-compressible boundary layer edge velocity ratio;a is the target parameter to be solved; and respectively represent the first auxiliary function and the second auxiliary function;n is the index of the viscous power law model;B and -A 2 respectively are the linear term and the quadratic term coefficients in the average temperature-velocity relationship.

4. The high-precision prediction method for high-speed turbulent boundary layer friction based on compressible transformation according to claim 2, characterized in that, The pre-calculation described in step S3 is as follows: First, parameters B and A are calculated 2 : ； ； In the formula: s is the Reynolds analogy factor, with a value of 1.14; Pr is the Prandtl number; T r To restore temperature; The wall temperature; Reference temperature for incoming flow; Then the calculated B and A 2 And the exponent n of the viscous power-law model is substituted into the first auxiliary function. The second auxiliary function is obtained by numerical integration or by analytical solution of the incomplete β function. Finally, the ratio of the free-flow velocity before and after the transformation is calculated. .

5. The high-precision prediction method for high-speed turbulent boundary layer friction based on compressible transformation according to claim 2, characterized in that, Step S4 uses Newton's iteration method for solving the problem. The specific steps are as follows: S41: Based on the current iteration value Calculate the integral variable; ； ； In the formula: and , respectively, are the first intermediate integral and the second intermediate integral; z is the dimensionless average velocity of the free flow velocity in the compressible case, and Z is the dimensionless average velocity of the free flow velocity in the equivalent incompressible case; S42: Calculate the Reynolds number for the current step's incompressible momentum thickness. The calculation formula is as follows: ； S43: Calculate the target value a target The calculation formula is as follows: ； In the formula: The equivalent logarithmic law slope constant is 0.

375. is the Kármán constant in the Coles-Fernholz formula, with a value of 0.384; This is the intercept constant in the Coles-Fernholz formula, with a value of 4.127; S44: Calculate the current iteration value and target value a target deviation If the deviation is less than the convergence criterion, then it is considered convergent, and the current iteration value is set. The objective parameter 'a' is used as the solution; otherwise, the deviation is calculated. about The derivative of the derivative is used, and the iteration value is updated using Newton's iteration method; Repeat steps S41 to S43 until convergence or the maximum number of iterations is reached.

6. The high-precision prediction method for high-speed turbulent boundary layer friction based on compressible transformation according to claim 2, characterized in that, The wall friction coefficient C in step S5 f The calculation formula is as follows: ； In the formula: a is the target parameter obtained by solving. The equivalent logarithmic law slope constant, For the incoming flow reference temperature, This refers to the wall temperature.

7. The high-precision prediction method for high-speed turbulent boundary layer friction based on compressible transformation according to claim 2, characterized in that, The heat flow coefficient B in step S5 q The calculation formula is as follows: ； In the formula: C f T is the wall friction coefficient; r To restore temperature, For the incoming flow reference temperature, Let be the wall temperature, and s be the Reynolds analogue factor.

8. The high-precision prediction method for high-speed turbulent boundary layer friction based on compressible transformation according to claim 1, characterized in that, The initial iteration value in step S2 is set to 10, and the convergence criterion is set to 10 -8 , and the maximum iteration step is set to 50.

9. An application of the prediction method according to any one of claims 1 to 7, characterized in that, After predicting the wall friction coefficient and heat flux coefficient, they are used for benchmark evaluation of aerodynamic friction and heat flux on the surface of high-speed aircraft to guide the preliminary design of aerodynamic shape and thermal protection system.

Images