Wall surface heating / cooling underwater curved surface boundary layer transition forecasting method

Abstract

The invention relates to a transition forecasting method of an underwater curved surface boundary layer for wall surface heating / cooling, which comprises the following steps of: acquiring an underwater curved surface appearance, and determining an underwater navigation working condition to be calculated; calculating a laminar flow basic flow field of the underwater curved surface boundary layer under the determined working condition; analyzing the linear flow stability of the elementary flow; in stability analysis, underwater wall surface heating / cooling is considered, a disturbance energy equation is established, and temperature disturbance, viscosity disturbance and heat conduction coefficient disturbance are introduced; the flow direction curvature and the circumferential curvature of the underwater curved surface appearance are considered at the same time; through linear stability analysis, growth rate distribution of small disturbances with different frequencies along the curved surface appearance is obtained; integrating small disturbance growth rates under different frequencies along the surface of the curved surface appearance, and obtaining an amplification factor N value; calculating the maximum envelope line of the amplification factor N value of different frequency waves along the surface of the curved surface appearance; and predicting the natural transition position of the underwater curved surface boundary layer of wall surface heating / cooling according to the experience transition threshold Ntr.

Description

Technical Field

[0001] The invention relates to the technical field of hydrodynamic research, and in particular to a method for predicting transition of an underwater curved surface boundary layer for wall heating / cooling. Background Art

[0002] The transition of flow state from laminar flow to turbulent flow is an important issue in fluid mechanics. Transition is of great significance to aerospace, underwater vehicle design and other fields. For underwater vehicles, transition has a great impact on flow-induced noise, which in turn affects the detection capability of the vehicle's bow sonar. Therefore, it is particularly important to find a suitable way to control the transition position. Wall heating / cooling is an important means to control the transition position. The shape of an underwater vehicle is generally a curved surface. Therefore, it is very important to accurately calculate the transition position of underwater curved surface flow with wall heating / cooling.

[0003] For the transition prediction problem of underwater curved surface boundary layer with wall heating / cooling, on the one hand, curved surface flow is much more complicated than plane flow; on the other hand, typical incompressible underwater flow only considers velocity field and pressure field, while wall heating / cooling introduces changes in temperature field, which makes the control equations and solution process more complicated, so there are few related research reports. It can be seen that the solution to this problem mainly lies in two aspects: one is to simulate the real underwater curved surface flow, that is, to consider the curvature change of the wall. There have been sufficient studies in this regard in recent years (Liu, J., Chu, X., and Zhang, Y.. Numerical investigation of natural transitions of bow boundary layers over underwater axisymmetric bodies. Phys. Fluids 33, 074101, 2021; Liu, J., Jianhua Liu, J., and Zhang, Y.. Influence of Reynolds number on the natural transition of boundary layers over underwater axisymmetric bodies. Phys. Fluids 35, 044107, 2023); the second is to simulate the underwater temperature field formed when the wall is heated / cooled. There has been work in this regard for flat-plate Blasius flow (Lv, J. and Zhang, Y.. Numerical study of the natural transition of water boundary layers over heated / cooled flat plates.Phys.Fluids35,064106,2023; Zhang Yongming, Lv Junbo. "A method for predicting the transition position of underwater flat plate boundary layer with wall heating / cooling.", CN202210174746.9.2022). However, the natural transition research that considers both underwater wall heating / cooling and wall curvature is only experimental research, and no numerical research has been seen so far. The transition experiment has high requirements on experimental conditions and is expensive.

[0004] In summary, it is necessary to explore a natural transition numerical prediction method that can take into account both wall heating / cooling and wall curvature. Summary of the invention

[0005] The object of the present invention is to provide a method for predicting the transition position of the underwater curved surface boundary layer of wall heating / cooling. The technical solution of the present invention is as follows:

[0006] A method for predicting transition of underwater curved surface boundary layer for wall heating / cooling comprises the following steps:

[0007] Step 1: Obtain the underwater curved surface shape and determine the underwater navigation conditions to be calculated, including the incoming flow velocity, the incoming flow temperature, and the wall temperature;

[0008] Step 2, calculating the laminar basic flow field of the underwater curved surface boundary layer with the working conditions determined in step 1: considering the underwater wall heating / cooling and the change of the wall curvature, the viscosity and thermal conductivity of water are regarded as functions of the flow field temperature, and considering the changes of the stream curvature and circumferential curvature of the curved surface shape, the laminar basic flow field is calculated by solving the Navier-Stokes (NS) equations;

[0009] Step three, based on the laminar basic flow field obtained in step two, analyze the flow stability of the underwater curved surface boundary layer laminar basic flow with wall heating / cooling; in the stability analysis, consider the underwater wall heating / cooling, establish the energy equation of the disturbance, introduce the temperature disturbance, viscosity disturbance and thermal conductivity disturbance; consider the flow curvature and circumferential curvature of the underwater curved surface shape at the same time; obtain the growth rate distribution of small disturbances of different frequencies along the curved surface shape through linear stability analysis;

[0010] Step 4, integrating the growth rate of small disturbances at different frequencies along the surface of the curved surface to obtain the value of the amplification factor N;

[0011] Step 5: Calculate the maximum envelope of the amplification factor N of waves of different frequencies along the curved surface, that is, N max curve;

[0012] Step 6: Determine the transition threshold N based on experience tr Predict the natural transition location of the underwater boundary layer for wall heating / cooling.

[0013] Furthermore, the method of step three is as follows:

[0014] (a) Establish the linear stability equation:

[0015] Based on the dimensionless NS equations in the body-fitted orthogonal curvilinear coordinate system, the NS equations (1) for underwater curved surface flow suitable for wall heating / cooling are established by considering the changes of water viscosity and thermal conductivity with temperature, and the flow direction and circumferential curvature of the wall:

[0016]

[0017] Where t is the time, q1, q2, and q3 are the streamwise, normal, and circumferential coordinates, respectively. is the instantaneous quantity vector, where and They are the stream velocity, normal velocity, circumferential velocity, pressure and temperature in the instantaneous quantities respectively. is the coefficient matrix of the linear terms in equation (1), is the nonlinear term in equation (1).

[0018] Decompose the instantaneous quantity of the flow field into the sum of the basic flow and the disturbance quantity, that is, Where Φ=(u,v,w,p,T) T is the basic flow vector, u, v, w, p and T are the stream velocity, normal velocity, circumferential velocity, pressure and temperature of the basic flow respectively, Φ'=(u',v',w',p',T') T is a small disturbance vector, u', v', w', p' and T' are the stream velocity disturbance, normal velocity disturbance, circumferential velocity disturbance, pressure disturbance and temperature disturbance respectively.

[0019] Both the instantaneous quantity and the basic flow satisfy the NS equation. The linear perturbation equation (2) is obtained by subtracting the basic flow equation from the instantaneous quantity equation and ignoring the nonlinear high-order small quantities:

[0020]

[0021] Among them, Γ, A, B, C, D, F 11 、F 22 、F 33 、F 12 、F 13 and F 23 is the coefficient matrix of equation (2), which contains the stream velocity u, normal velocity v, circumferential velocity w, pressure p, temperature T, dynamic viscosity μ and thermal conductivity k in the basic flow, and also contains the Reynolds number Re, Prandtl number Pr and Eckert number Ec.

[0022] All coefficient matrices in the linear perturbation equation (2) are square matrices of order 5. The coefficient matrix Γ is a diagonal matrix, which can be expressed as Γ = diag(1,1,1,1,0).

[0023] The non-zero elements in the coefficient matrices A, B, and C are:

[0024]

[0025]

[0026] Among them, h1=1+q2×drx, h2=1, h3=1+q2×drz are the Lame coefficients in the stream direction, normal direction and circumferential direction respectively, and drx and drz are the stream curvature and circumferential curvature of the wall respectively. In order to simplify the equation, an intermediate variable is introduced: In addition, μ T and kT They are

[0027] The non-zero elements in the coefficient matrix D are:

[0028]

[0029] Among them, the intermediate variables M, s 122 ,s 322 , d 122 , d 322 and d 1322 They are defined as

[0030]

[0031] Coefficient matrix F 11 、F 22 and F 33 They are all diagonal matrices and can be expressed as:

[0032]

[0034] Coefficient matrix F 12 、F 13 and F 23 are all symmetric matrices with non-zero elements:

[0035]

[0036] After the parallel flow assumption is introduced, the disturbance is set to the traveling wave form This leads to the linear stability equation (3):

[0037]

[0038] No-slip boundary conditions and isothermal boundary conditions (4) are used at the wall:

[0039]

[0040] The amplitude of the small disturbance wave in the far field tends to zero, and the far field boundary condition (5) is:

[0041]

[0042] Where α, β and ω are the streamwise wave number, circumferential wave number and circular frequency respectively, and cc is the complex conjugate. In equation (3), is the characteristic function vector of the small perturbation, where They are the characteristic functions of stream velocity, normal velocity, circumferential velocity, pressure and temperature; All are coefficient matrices and can be calculated according to formula (6); the quantities in equations (1), (2) and (3), formulas (4) and (5) are dimensionless quantities without the superscript "*", while the dimensional quantities in this patent have the superscript "*";

[0043]

[0044] (b) Numerical solution:

[0045] The linear stability equation (3) is solved numerically to obtain the characteristic relationship between the small disturbance frequency, wave number and flow field Reynolds number at different flow direction positions when considering wall heating / cooling and wall curvature, that is, the neutral curve and the growth rate of different frequencies (-α i * )Cloud map.

[0046] Furthermore, the method of step 4 is as follows:

[0047] For a small disturbance of a given frequency, starting from entering the growth zone, the amplitude growth rate is integrated along the surface of the curved shape to obtain the distribution of the amplification factor N values ​​of small disturbances of different frequencies along the flow direction. The integral formula is given by (7):

[0048]

[0049] Among them, f * is the frequency of the small disturbance, q1 * is a certain flow direction position, q0 * is the flow direction into the growth zone, -α i * is the growth rate of small disturbances.

[0050] Furthermore, the method of step five is as follows:

[0051] Take the maximum value of the amplification factor N along the streamwise coordinate of small disturbances of different frequencies and take the envelope to get N max Curve, the calculation formula is given by (8):

[0052]

[0053] Furthermore, the method of step six is ​​as follows:

[0054] For natural transition, it is believed that as long as the small disturbance wave of a certain frequency reaches the initial amplitude during the process of propagating downstream, times, the laminar flow develops into turbulent flow; the N obtained in step 5 max Find N on the curve tr The position at which the natural transition begins is predicted for the underwater curved surface boundary layer as wall heating / cooling, corresponding to the streamwise position.

[0055] The substantial difference between the present invention and the prior art (patent CN202210174746.9) is that the underwater wall heating / cooling and the change of the wall curvature of the curved surface shape are considered simultaneously in the basic flow calculation, and the underwater wall heating / cooling and the change of the wall curvature are also considered simultaneously in the stability analysis, and the stability equation of the underwater curved surface boundary layer suitable for wall heating / cooling is re-derived. Advantageous effects compared with the prior art:

[0056] (1) The present invention establishes a stability analysis method for the underwater boundary layer that can simultaneously consider the wall curvature change and wall heating / cooling.

[0057] (2) The present invention provides a feasible and reasonable method for predicting the natural transition position of the underwater curved surface boundary layer during wall heating / cooling.

[0058] (3) The present invention can provide theoretical support for realizing transition control of the bow of an underwater vehicle, thereby achieving drag reduction and noise reduction. BRIEF DESCRIPTION OF THE DRAWINGS

[0059] Figure 1 Schematic diagram of the steps of the present invention

[0060] Figure 2 Neutral curves for three different wall temperature conditions

[0061] Figure 3 The growth rate distribution cloud diagram for three different wall temperature conditions

[0062] Figure 4 The N value distribution cloud diagram for three different wall temperature conditions

[0063] Figure 5 is N for three different wall temperature conditions max curve

[0064] Figure 6 Comparison of the results of this method and the experimentally measured transition Reynolds number DETAILED DESCRIPTION

[0065] The present invention is described in detail below in conjunction with specific embodiments and accompanying drawings. In the following description, for the purpose of explanation and not limitation, specific details are set forth to help fully understand the present invention. However, it is apparent to those skilled in the art that the present invention may also be practiced in other embodiments that depart from these specific details.

[0066] It should be noted that, in order to avoid obscuring the present invention due to unnecessary details, only the device structure and / or processing steps closely related to the solution according to the present invention are shown in the accompanying drawings, while other details that are not closely related to the present invention are omitted.

[0067] The embodiment of the present invention provides a method for predicting the transition position of the underwater curved surface boundary layer of wall heating / cooling, such as Figure 1 As shown, the following steps are included:

[0068] Step 1: Provide the underwater surface shape and determine the actual underwater navigation conditions to be calculated, namely, the incoming flow velocity, the incoming flow temperature, and the wall temperature.

[0069] In this step, since underwater vehicles mostly use a body of revolution shape, the curved surface shape of the underwater vehicle adopts the body of revolution shape used in the literature (Lauchle, GC and Gurney, GB. Laminar boundary-layer transition on aheated underwater body. J. Fluid Mech. 1984, 144: 79–101.), and the angle of attack of the body of revolution in this embodiment is 0°.

[0070] In this step, we first calculate three The working conditions are 10.0℃ (wall cooling), 26.7℃ (wall temperature equals to the incoming flow temperature), and 46.4℃ (wall heating). The incoming flow velocities of these three working conditions are The incoming flow temperature is This embodiment will give the neutral curve, growth rate cloud diagram, N value cloud diagram, N value cloud diagram of these three working conditions. max Curves and turning points.

[0071] In addition, the present invention also provides a comparison with the transition results of experimental measurements. Four wall heating conditions with different wall temperatures are selected, and the specific conditions are shown in Table 1, where the temperature difference is the difference between the wall temperature and the incoming flow temperature.

[0072] Table 1 Four wall heating conditions compared with the experiment

[0073] Flow velocity (m / s) 8.60 12.85 14.03 14.17 Inflow temperature / ℃ 25.6 26.7 27.2 28.3 Wall temperature / ℃ 41.1 46.4 49.2 51.8 Temperature difference / ℃ 15.5 19.7 22.0 23.5

[0074] Step 2: Calculate the laminar basic flow field of the underwater curved surface boundary layer with the working conditions determined in step 1. When solving the laminar basic flow, both the underwater wall heating / cooling and the change of the wall curvature must be considered. Specifically, the viscosity and thermal conductivity of water are considered as functions of the flow field temperature; at the same time, the flow curvature and circumferential curvature of the curved surface are considered, and the laminar basic flow field is calculated by solving the Navier-Stokes (NS) equations.

[0075] In this step, because the underwater curved surface is a body of revolution and the calculated attack angle is 0°, it can be simplified to a two-dimensional flow field calculation.

[0076] In this step, the basic flow is numerically solved using Ansys Fluent software. The user-defined function (UDF) in the software is used to define the functional relationship between the viscosity and thermal conductivity of water and temperature, and the functional relationship uses the expression provided by the literature (see Lv, J. and Zhang, Y.. Numerical study of the natural transition of water boundary layers over heated / cooled flat plates. Phys. Fluids 35, 064106, 2023). The wall uses the boundary condition of constant wall temperature and turns on the energy equation to simulate the basic flow field of laminar flow when the wall is heated / cooled.

[0077] Step 3: Based on the linear stability theory, analyze the flow stability of each laminar basic flow in step 2:

[0078] Based on the laminar basic flow field obtained in step 2, the flow stability of the laminar basic flow of the underwater curved surface boundary layer with wall heating / cooling is analyzed. In the stability analysis, it is necessary to consider the underwater wall heating / cooling, that is, consider the energy equation of the disturbance, introduce temperature disturbance, viscosity disturbance and thermal conductivity disturbance; at the same time, consider the flow curvature and circumferential curvature of the underwater curved surface shape. Through linear stability analysis, the growth rate distribution of small disturbances of different frequencies along the curved surface shape can be obtained as follows:

[0079] (1) Establish the linear stability equation:

[0080] Starting from the dimensionless NS equations in a body-fitted orthogonal curvilinear coordinate system, the NS equations (1) for underwater curved surface flows applicable to wall heating / cooling are derived by considering the temperature variations of water viscosity and thermal conductivity as well as the wall curvature.

[0081]

[0082] Where t is the time, q1, q2, and q3 are the streamwise, normal, and circumferential coordinates, respectively. is the instantaneous quantity vector, where and They are the stream velocity, normal velocity, circumferential velocity, pressure and temperature in the instantaneous quantities respectively. is the coefficient matrix of the linear terms in equation (1), is the nonlinear term in equation (1).

[0083] Decompose the instantaneous quantity of the flow field into the sum of the basic flow and the disturbance quantity, that is, Where Φ=(u,v,w,p,T) Tis the basic flow vector, u, v, w, p and T are the stream velocity, normal velocity, circumferential velocity, pressure and temperature of the basic flow respectively, Φ'=(u',v',w',p',T') T is a small disturbance vector, u', v', w', p' and T' are the stream velocity disturbance, normal velocity disturbance, circumferential velocity disturbance, pressure disturbance and temperature disturbance respectively.

[0084] Both the instantaneous quantity and the basic flow satisfy the NS equation. The linear perturbation equation (2) is obtained by subtracting the basic flow equation from the instantaneous quantity equation and ignoring the nonlinear high-order small quantities:

[0085]

[0086] Among them, Γ, A, B, C, D, F 11 、F 22 、F 33 、F 12 、F 13 and F 23 is the coefficient matrix of equation (2), which contains the stream velocity u, normal velocity v, circumferential velocity w, pressure p, temperature T, dynamic viscosity μ and thermal conductivity k in the basic flow, and also contains the Reynolds number Re, Prandtl number Pr and Eckert number Ec.

[0087] All coefficient matrices in the linear perturbation equation (2) are square matrices of order 5. The coefficient matrix Γ is a diagonal matrix, which can be expressed as Γ = diag(1,1,1,1,0).

[0088] The non-zero elements in the coefficient matrices A, B, and C are:

[0089]

[0091]

[0092] Among them, h1=1+q2×drx, h2=1, h3=1+q2×drz are the Lame coefficients in the stream direction, normal direction and circumferential direction respectively, and drx and drz are the stream curvature and circumferential curvature of the wall respectively. In order to simplify the equation, an intermediate variable is introduced: In addition, μ T and k T They are

[0093] The non-zero elements in the coefficient matrix D are:

[0094]

[0096]

[0097] Among them, the intermediate variables M, s 122 ,s 322 , d 122 , d 322 and d 1322 They are defined as

[0098] Coefficient matrix F 11 、F 22 and F 33 They are all diagonal matrices and can be expressed as:

[0099]

[0100] Coefficient matrix F 12 、F 13 and F 23 are all symmetric matrices with non-zero elements:

[0101]

[0102] After the parallel flow assumption is introduced, the disturbance is set to the traveling wave form This leads to the linear stability equation (3):

[0103]

[0104] No-slip boundary conditions and isothermal boundary conditions (4) are used at the wall:

[0105]

[0106] The amplitude of the small disturbance wave in the far field tends to zero, and the far field boundary condition (5) is:

[0107]

[0108] Where α, β and ω are the streamwise wave number, circumferential wave number and circular frequency respectively, and cc is the complex conjugate. In equation (3), is the characteristic function vector of the small perturbation, where They are the characteristic functions of stream velocity, normal velocity, circumferential velocity, pressure and temperature; is the coefficient matrix, which can be calculated according to formula (6); the quantities in equations (1), (2) and (3), formulas (4) and (5) are dimensionless quantities without the superscript "*", while the dimensional quantities in this patent have the superscript "*";

[0109]

[0110] (2) Numerical solution:

[0111] By using numerical methods to solve the linear stability equation (3) in spatial mode, we can obtain the characteristic relationship between the small disturbance frequency, wave number and flow field Reynolds number at different flow direction positions when considering wall heating / cooling and wall curvature, that is, the neutral curve and the growth rate of different frequencies (-α i * )Cloud map.

[0112] In this step, the numerical calculation method of the characteristic value of the stability equation can adopt means known in the art (for details, see "Flow Stability", Zhou Heng et al., National Defense Industry Press).

[0113] In this step, the neutral curve is the dividing line between the stable region and the unstable region, which is a well-known technique in the art (for details, see "Flow Stability", Zhou Heng et al., National Defense Industry Press).

[0114] The neutral curves of the three different wall temperature conditions obtained in this step are as follows: Figure 2 As shown; the distribution cloud diagram of the frequency amplitude growth rate of the three working conditions along the flow direction, that is, the growth rate distribution cloud diagram is as follows Figure 3 shown.

[0115] Step 4: Based on the results of the stability analysis in step 3, the growth rate of small disturbances at different frequencies is integrated along the surface of the curved shape to obtain the amplification factor N value. The method is as follows:

[0116] For a small disturbance of a given frequency, starting from entering the growth zone, the amplitude growth rate is integrated along the surface of the curved shape, and the distribution of the amplification factor N values ​​of small disturbances of different frequencies along the flow direction can be obtained. The integral formula is given by (7):

[0117]

[0118] Among them, f * is the frequency of the small disturbance, q1 * is a certain flow direction position, q0 * is the flow direction into the growth zone, -α i * is the growth rate of small disturbances.

[0119] In this step, the distribution of the amplification factor N value of each frequency small disturbance along the flow direction, that is, the N value distribution cloud diagram of three different wall temperature conditions is as follows: Figure 4 shown.

[0120] Step 5: Calculate the maximum envelope of the amplification factor N of the surface of the curved surface with different frequency waves, that is, N max Curve, the method is as follows:

[0121] Take the maximum value of the amplification factor N along the streamwise coordinate of small disturbances of different frequencies and take the envelope to get Nmax Curve, the calculation formula is given by (8):

[0122]

[0123] In this step, N of three different wall temperature conditions max Curve like Figure 5 shown.

[0124] Step 6: Determine the transition threshold N based on experience tr Predict the natural transition location of the underwater curved surface boundary layer for wall heating / cooling as follows:

[0125] For natural transition, it is believed that as long as the small disturbance wave of a certain frequency reaches the initial amplitude during the process of propagating downstream, times, the laminar flow develops into turbulent flow, so N tr is the criterion for predicting the natural transition position in the present invention; N obtained in step 5 max Find N on the curve tr The position at which the natural transition begins is predicted for the underwater curved surface boundary layer as wall heating / cooling, corresponding to the streamwise position.

[0126] In this step, the empirical transition threshold N recommended by the literature (Lv, J. and Zhang, Y.. Numerical study of the natural transition of water boundary layers over heated / cooled flatplates. Phys. Fluids 35, 064106, 2023) is used. tr =7.

[0127] In this step, according to the empirical transition threshold, the N obtained in step 5 can be max The transition flow direction position found on the curve is the predicted transition position, as shown in Table 2.

[0128] Table 2 Transition positions of three different wall temperature conditions

[0129]

[0130] In this step, in order to compare with the experimental results, we calculated the transition positions of the four working conditions in the experiment and gave them in the form of Reynolds number. The comparison results are shown in Figure 6 As shown, the comparison results are in good agreement, where the transition Reynolds number is defined as is the kinematic viscosity of the incoming flow, It is a turning point.

[0131] Features described and / or illustrated above for one embodiment may be used in the same or similar manner in one or more other embodiments, and / or combined with or used in place of features in other embodiments.

[0132] It should be emphasized that the term "include / comprises" when used herein refers to the presence of features, integers, steps or components, but does not exclude the presence or addition of one or more other features, integers, steps, components or combinations thereof.

[0133] The above devices and methods of the present invention can be implemented by hardware, or by hardware combined with software. The present invention relates to such a computer-readable program, which, when executed by a logic component, enables the logic component to implement the above-mentioned devices or components, or enables the logic component to implement the above-mentioned various methods or steps. The present invention also relates to a storage medium for storing the above-mentioned program, such as a hard disk, a magnetic disk, an optical disk, a DVD, a flash memory, etc.

[0134] The many features and advantages of these embodiments are apparent from this detailed description, and thus the appended claims are intended to cover all such features and advantages of these embodiments that fall within their true spirit and scope. Furthermore, since numerous modifications and changes will readily occur to those skilled in the art, it is not intended that the embodiments of the invention be limited to the exact construction and operation illustrated and described, but rather all suitable modifications and equivalents falling within the scope thereof are intended to be covered.

[0135] Parts of the present invention that are not described in detail are well known to those skilled in the art.

Claims

1. A method for predicting the transition of an underwater curved surface boundary layer with wall heating / cooling, comprising the following steps: Step 1: Obtain the underwater curved surface shape and determine the underwater navigation conditions to be calculated, including the incoming flow velocity, the incoming flow temperature, and the wall temperature; Step 2, calculating the laminar basic flow field of the underwater curved surface boundary layer with the working conditions determined in step 1: considering the underwater wall heating / cooling and the change of the wall curvature, the viscosity and thermal conductivity of water are regarded as functions of the flow field temperature, and considering the changes of the stream curvature and circumferential curvature of the curved surface shape, the laminar basic flow field is calculated by solving the Navier-Stokes (NS) equations; Step three, based on the laminar basic flow field obtained in step two, analyze the flow stability of the underwater curved surface boundary layer laminar basic flow with wall heating / cooling; in the stability analysis, consider the underwater wall heating / cooling, establish the energy equation of the disturbance, introduce the temperature disturbance, viscosity disturbance and thermal conductivity disturbance; consider the flow curvature and circumferential curvature of the underwater curved surface shape at the same time; through linear stability analysis, obtain the growth rate distribution of small disturbances of different frequencies along the curved surface shape; the method is as follows: (a) Establish the linear stability equation: Based on the dimensionless NS equations in the body-fitting orthogonal curvilinear coordinate system, the NS equations for underwater curved surface flow suitable for wall heating / cooling are established by considering the changes of water viscosity and thermal conductivity with temperature, and the flow direction and circumferential curvature of the wall. The instantaneous quantity of the flow field is decomposed into the sum of the basic flow and the disturbance quantity. Both the instantaneous quantity and the basic flow satisfy the NS equations. The equation of the basic flow is subtracted from the equation of the instantaneous quantity, and the nonlinear high-order small quantities are ignored to obtain the linear disturbance equation (1): Where t is time, q1 is the streamwise coordinate, q2 is the normal coordinate, and q3 is the circumferential coordinate; Φ' = (u', v', w', p', T') T is a small disturbance vector, u', v', w', p' and T' are the flow velocity disturbance, normal velocity disturbance, circumferential velocity disturbance, pressure disturbance and temperature disturbance respectively; Γ, A, B, C, D, F 11 、F 22 、F 33 、F 12 、F 13 and F 23 is the coefficient matrix of the linear perturbation equation (1), which contains the stream velocity u, normal velocity v, circumferential velocity w, pressure p, temperature T, dynamic viscosity μ and thermal conductivity k in the basic flow, and also contains the Reynolds number Re, Prandtl number Pr and Eckert number Ec; All coefficient matrices in the linear perturbation equation (1) are 5-order square matrices; the coefficient matrix Γ is a diagonal matrix, expressed as Γ = diag(1,1,1,1,0); The non-zero elements in the coefficient matrices A, B, and C are: Among them, h1=1+q2×drx, h2=1, h3=1+q2×drz are the Lame coefficients in the stream direction, normal direction and circumferential direction respectively, drx and drz are the stream curvature and circumferential curvature of the wall respectively; in order to simplify the equation, the intermediate variable is introduced: In addition, μ T and k T They are The non-zero elements in the coefficient matrix D are: Among them, the intermediate variables M, s 122 ,s 322 , d 122 , d 322 and d 1322 They are defined as Coefficient matrix F 11 、F 22 and F 33 They are all diagonal matrices, expressed as: Coefficient matrix F 12 、F 13 and F 23 are all symmetric matrices with non-zero elements: After the parallel flow assumption is introduced, the disturbance is set to the traveling wave form The linear stability equation (2) is obtained: No-slip boundary conditions and isothermal boundary conditions (3) are used at the wall: The amplitude of the small disturbance wave in the far field tends to zero, and the far field boundary condition (4) is: Where α, β and ω are the streamwise wave number, circumferential wave number and circular frequency respectively, and cc is the complex conjugate; in the linear stability equation (2), is the characteristic function vector of the small perturbation, where They are the characteristic functions of stream velocity, normal velocity, circumferential velocity, pressure and temperature; All are coefficient matrices, calculated according to formula (5); the quantities in equations (1) and (2), formulas (3) and (4) are dimensionless quantities without "*" superscripts, while dimensional quantities have "*" superscripts; (b) Numerical solution: The linear stability equation (2) is solved numerically to obtain the characteristic relationship between the small disturbance frequency, wave number and flow field Reynolds number at different flow direction positions when considering wall heating / cooling and wall curvature, that is, the neutral curve and the growth rate of different frequencies (-α i * ) Cloud map; Step 4, integrating the growth rate of small disturbances at different frequencies along the surface of the curved surface to obtain the value of the amplification factor N; Step 5: Calculate the maximum envelope of the amplification factor N of waves of different frequencies along the curved surface, that is, N max curve; Step 6: Determine the transition threshold N based on experience tr Predict the natural transition location of the underwater boundary layer for wall heating / cooling.

2. The method for predicting transition of underwater curved surface boundary layer for wall heating / cooling according to claim 1, characterized in that: The method for step 4 is as follows: For a small disturbance of a given frequency, starting from entering the growth zone, the amplitude growth rate is integrated along the surface of the curved shape to obtain the distribution of the amplification factor N values ​​of small disturbances of different frequencies along the flow direction. The integral formula is given by (6): Among them, f * is the frequency of the small disturbance, q1 * is a certain flow direction position, q0 * is the flow direction into the growth zone, -α i * is the growth rate of small disturbances.

3. The method for predicting transition of underwater curved surface boundary layer for wall heating / cooling according to claim 2, characterized in that: The method for step five is as follows: Take the maximum value of the amplification factor N along the streamwise coordinate of small disturbances of different frequencies and obtain N max Curve, the calculation formula is given by (7):

4. The method for predicting transition of underwater curved surface boundary layer for wall heating / cooling according to claim 3, characterized in that: The method for step six is ​​as follows: For natural transition, it is believed that as long as the small disturbance wave of a certain frequency reaches the initial amplitude during the process of propagating downstream, times, the laminar flow develops into turbulent flow; the N obtained in step 5 max Find N on the curve tr The position at which the natural transition begins is predicted for the underwater curved surface boundary layer as wall heating / cooling, corresponding to the streamwise position.

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