A method for simulating flow of a liquid film at small reynolds numbers

Abstract

The application discloses a small Reynolds number liquid film flow simulation method, which is based on a flow physical model, derives normal stress and shear stress balance conditions and motion boundary conditions of a liquid film surface under the action of gravity, airflow shear force and evaporation effect, and carries out non-dimensionalization and simplification on wall surface non-slip boundary conditions, flow mass equation and momentum equation and liquid film surface boundary conditions; aiming at small Reynolds number flow, the method is based on long wave theory to derive a differential equation of liquid film thickness evolution with time and space under the synergistic influence of gravity, airflow shear force and evaporation effect; the differential equation is subjected to weak nonlinear evolution analysis of wave, the change relationship of wave instability growth rate with gravity, airflow shear force and evaporation effect is obtained, and a change image of the influence of airflow shear force and evaporation effect on the liquid film wave instability growth rate is given.

Description

TECHNICAL FIELD

[0001] The application belongs to the technical field of fluid mechanics, and particularly relates to a small Reynolds number liquid film flow simulation method. BACKGROUND

[0002] Liquid film flow phenomena exist widely in daily life, production and scientific and technological fields, such as in the biological field, including the flow of human tissue fluid and the industrial field. Liquid film flow phenomena also exist in the field of aircraft thermal protection. For silicon-based materials, it is a heat-proof material mainly changed by molten flow, and under the condition of aerodynamic heating, the solid material will melt to form a molten state and flow under the action of air shear force and gravity.

[0003] The research on liquid film flow phenomena started in the 1950s and 1960s of the last century. Kapitza and his son first discovered the liquid film flow phenomenon in experiments and conducted theoretical research. Since then, the subject of liquid film flow has attracted the attention of many scholars. They use boundary layer integration method, long wave theory method and other methods based on control equation, constitutive equation and boundary condition to study the liquid film flow phenomenon, but the above researches are still limited to the study of liquid film flow and wave transmission under the action of gravity. For the study of liquid film flow under the action of air shear force and evaporation effect, there are mostly only experimental tests, such as Budakli who used experimental method to study the flow and heat transfer phenomenon of liquid film under the action of gravity and air shear force. At present, there is a lack of theoretical and numerical research on liquid film flow and wave transmission under the action of air shear force and evaporation effect. SUMMARY

[0004] The technical problem of the application is to overcome the shortcomings of the prior art and provide a small Reynolds number liquid film flow simulation method, which solves the problem that the liquid film flow under the influence of gravity-air shear force-evaporation effect cannot be studied simultaneously in the prior art.

[0005] In order to solve the above technical problems, the application discloses a small Reynolds number liquid film flow simulation method, which comprises the following steps:

[0006] S1, dimensionless processing of parameters is performed to obtain dimensionless parameters;

[0007] S2, for the flow of liquid film over an inclined rigid plate under the action of gravity, air shear force and evaporation effect, corresponding mass equation a and momentum equation a are given;

[0008] S3, for the liquid film surface under the action of gravity, air shear force, gas pressure and evaporation effect, corresponding normal stress balance condition a, shear stress balance condition a, wall no-slip boundary condition a and liquid film surface motion boundary condition a under the influence of evaporation effect are given;

[0009] S4, based on the dimensionless parameter, the mass equation a, momentum equation a, normal stress balance condition a, shear stress balance condition a, wall no-slip boundary condition a and liquid film surface movement boundary condition a are respectively dimensionless processed to obtain dimensionless mass equation b, momentum equation b, normal stress balance condition b, shear stress balance condition b, wall no-slip boundary condition b and liquid film surface movement boundary condition b;

[0010] S5, the Weber number We is scaled to make the scaled Weber number of the order of 1, the scaled Weber number is respectively substituted into the normal stress balance condition b, and small quantities are omitted by magnitude analysis to obtain the normal stress balance condition c;

[0011] S6, the dimensionless liquid film velocity along the flow direction, the normal velocity of the liquid film and the pressure in the liquid film are respectively expanded with a small quantity ε, the zero-order and first-order quantities are retained, substituted into the mass equation b, momentum equation b, normal stress balance condition c, shear stress balance condition b, wall no-slip boundary condition b and liquid film surface movement boundary condition b, and simplified to obtain the zero-order solution and first-order solution of the liquid film velocity along the flow direction; wherein, ε<<1;

[0012] S7, the mass equation b is integrated along the thickness direction of the liquid film, combined with the wall no-slip boundary condition b and the liquid film surface movement boundary condition b, and the zero-order solution and first-order solution of the liquid film velocity along the flow direction obtained in step S6 are substituted into to obtain a differential equation about the evolution of the liquid film thickness;

[0013] S8, based on the differential equation, wave weak nonlinear evolution analysis is carried out to obtain the variation law of the disturbance instability growth rate with the inclination angle, evaporation effect, gas flow shear force, Reynolds number and Weber number;

[0014] S9, the size of the evaporation effect and the gas flow shear force in step S8 is changed to obtain the variation curve of the instability growth rate, and the influence law of the evaporation effect and the gas flow shear force on the liquid film wave transmission instability growth rate is determined.

[0015] In the above small Reynolds number liquid film flow simulation method, the parameters are dimensionless processed to obtain dimensionless parameters, including:

[0016] A small quantity ε=h0 / l is introduced; wherein, h0 represents the initial liquid film thickness, and l represents the typical liquid-gas interface deformation wavelength;

[0017] The velocity of the liquid layer surface under no disturbance is introduced Wherein, ρ represents the density of the fluid in the liquid film, g represents the acceleration of gravity, and μ represents the viscosity at zero shear rate;

[0018] The Reynolds number Re=ρU0h0 / μ is introduced;

[0019] Weber number wherein γ0 represents a liquid film surface tension coefficient;

[0020] Based on ε, U0, Re and We, the liquid film flow direction coordinate x, the thickness direction coordinate y, the liquid film thickness h, the gas flow shear force τ g , the liquid film internal pressure p, the external gas pressure p g , the liquid film flow direction velocity u, the liquid film normal velocity v, the time t, the gas flow velocity U g and the liquid film surface velocity u s are dimensionally processed to obtain dimensionless parameters:

[0021]

[0022] wherein, U gg and are dimensionless parameters corresponding to x, y, h, σ xx , σ yy , σ xy , τ g , p, p g , u, v, t, U g and u s respectively.

[0023] In the above liquid film flow simulation method at small Reynolds number,

[0024] The expression of the mass equation a is as follows:

[0025]

[0026] The expression of the momentum equation a is as follows:

[0027]

[0028] wherein θ represents the included angle between the liquid film flow direction and the horizontal direction;

[0029] The expression of the normal stress balance condition a is as follows:

[0030]

[0031] wherein J represents the mass jump flow of the liquid film due to the interface phase change; h t represents the change rate of the liquid film thickness with respect to the time t, h x represents the change rate of the liquid film thickness with respect to x, h xx represents the second partial derivative of the liquid film thickness with respect to x, u x represents the partial derivative of the liquid film flow direction velocity with respect to x, u y denotes the partial derivative of the liquid film velocity along the flow direction with respect to y,

[0032] The expression of the shear stress balance condition a is as follows:

[0033]

[0034] The expression of the wall no-slip boundary condition a is as follows:

[0035]

[0036] The expression of the liquid film surface motion boundary condition a is as follows:

[0037]

[0038] where h(x, t) represents the rate of change of the liquid film thickness along the flow direction with time.

[0039] In the above liquid film flow simulation method of small Reynolds number, the original expression of J is as follows:

[0040]

[0041] ξ = 1 / (Ku·Pr), Ku = ΔH / (C p ·ΔT), Pr = v / a

[0042] where k represents the thermal conductivity, ΔH represents the latent heat of vaporization of the liquid, T represents the temperature in the liquid film, Ku represents the Kutateladzes number, C p represents the specific heat capacity of the fluid in the liquid film, ΔT represents the change of the temperature in the liquid film, Pr represents the Prandtl number, v represents the kinematic viscosity coefficient, a represents the thermal diffusion coefficient; ξ represents the mass jump flow rate, and when ξ takes a positive value, it represents the condensation state, and when ξ takes a negative value, it represents the evaporation state;

[0043] For the material aerodynamic heating surface, there is:

[0044]

[0045] Then, the expression of the simplified J is as follows:

[0046]

[0047] where q or represents the cold wall heat flux density.

[0048] In the above liquid film flow simulation method of small Reynolds number,

[0049] The expression of the mass equation b is as follows:

[0050]

[0051] The expression of the momentum equation b is as follows:

[0052]

[0053] The expression of the normal stress balance condition b is as follows:

[0054]

[0055] The expression of the shear stress balance condition b is as follows:

[0056]

[0057] The expression of the wall no-slip boundary condition b is as follows:

[0058]

[0059] The expression of the liquid film surface motion boundary condition b is as follows:

[0060]

[0061] wherein J A and J B are evaporation effect parameters, In the above liquid film flow simulation method of small Reynolds number, the Weber number We is scaled by the following way:

[0062]

[0063] wherein represents the scaled Weber number, and O(1) represents the order of 1.

[0064] In the above liquid film flow simulation method of small Reynolds number, the expression of the normal stress balance condition c is as follows:

[0065]

[0066] In the above liquid film flow simulation method of small Reynolds number, the dimensionless liquid film velocity along the flow direction, the normal velocity of the liquid film and the pressure in the liquid film are respectively expanded by a small quantity ε, and after the zero-order and first-order quantities are reserved, they are substituted into the mass equation b, the momentum equation b, the normal stress balance condition c, the shear stress balance condition b, the wall no-slip boundary condition b and the liquid film surface motion boundary condition b, and simplified to obtain the zero-order solution and the first-order solution of the liquid film velocity along the flow direction, including:

[0067] The dimensionless liquid film velocity along the flow direction, the normal velocity of the liquid film and the pressure in the liquid film are respectively expanded by a small quantity ε:

[0068]

[0069] wherein, denotes a dimensionless zeroth order liquid film velocity along the flow direction, a normal velocity of the liquid film, and a pressure inside the liquid film, denotes a dimensionless first order liquid film velocity along the flow direction, a normal velocity of the liquid film, and a pressure inside the liquid film, O(ε 2 ) denotes a small quantity of order ε 2 ;

[0070] Substituting and into the mass equation b, the momentum equation b, the normal stress balance condition c, the shear stress balance condition b, the wall no-slip boundary condition b, and the liquid film surface motion boundary condition b, the following simplification process is performed:

[0071]

[0072] wherein, denotes a differential of with respect to , denotes a differential of with respect to , denotes a differential of with respect to , denotes a differential of with respect to , denotes a differential of with respect to , denotes a differential of with respect to , denotes a differential of with respect to , denotes a differential of with respect to ;

[0073] Based on the equation (15), the zeroth order solution of the liquid film velocity along the flow direction and the first order solution are obtained.

[0074]

[0075] wherein, denotes a differential of with respect to , denotes a mixed differential of with respect to , denotes a differential of with respect to The third differential.

[0076] In the above-mentioned low Reynolds number liquid film flow simulation method, the mass equation b is integrated along the liquid film thickness direction. Combining the wall no-slip boundary condition b and the liquid film surface motion boundary condition b, and substituting the zero-order and first-order solutions of the liquid film velocity along the flow direction obtained in step S6, the differential equations concerning the evolution of the liquid film thickness are obtained, including:

[0077] Integrating mass equation b along the liquid film thickness direction, we get:

[0078]

[0079] q represents flow rate:

[0080]

[0081] Substituting the zero-order and first-order solutions of the liquid film velocity along the flow direction obtained in step S6 into formula (18), we get:

[0082]

[0083] Substituting equation (19) into equation (17), we obtain the differential equation for the evolution of the liquid film thickness:

[0084]

[0085] in, express right The fourth differential.

[0086] In the above-mentioned low Reynolds number liquid film flow simulation method, based on differential equations, a weakly nonlinear evolution analysis of the fluctuation is performed to obtain the variation law of the growth rate of disturbance instability with tilt angle, evaporation effect, airflow shear force, Reynolds number and Weber number, including:

[0087] make Substituting into the differential equation, neglecting O(ε) 2 )get:

[0088]

[0089] Where η represents the thickness of the liquid film after the substitution. Indicates η pairs The differential, Indicates η pairs The differential, Indicates η pairs The second differential, Indicates η pairs The fourth differential;

[0090] Let denotes the dimensionless film flow direction coordinate after substitution, denotes the dimensionless time after substitution, then:

[0091]

[0092] wherein, denotes the differential of η with respect to , denotes the differential of η with respect to , denotes the second differential of η with respect to , denotes the fourth differential of η with respect to ;

[0093] Considering the flat plate solution, i.e. η≡0, we have:

[0094]

[0095] Given: wherein, H<<1, i represents an imaginary unit, k represents a wave number, and ω represents an angular frequency;

[0096] then we have:

[0097]

[0098] then, the real part of ω is:

[0099]

[0100] wherein, ω r represents the real part of ω, and is used to represent the instability growth rate.

[0101] The present application has the following advantages:

[0102] Current calculation methods of liquid film flow only consider the flow under the action of gravity through vertical or inclined rigid or flexible flat plates, pipes and the like, and the small Reynolds number liquid film flow problem under the synergistic influence of gravity, air flow shear force and evaporation effect is still lacking in theoretical and numerical research. The present application takes aircraft thermal protection as the background, considers the small Reynolds number liquid film flow rule under the synergistic influence of gravity, air flow shear force and evaporation effect, improves the traditional long wave theory research method to consider more factors affecting the liquid film flow, and has the ability to analyze the small Reynolds number liquid film flow under the synergistic influence of gravity, air flow shear force and evaporation effect. BRIEF DESCRIPTION OF DRAWINGS

[0103] Figure 1 is a flow chart of a small Reynolds number liquid film flow simulation method in the embodiment of the present application;

[0104] Figure 2 is a physical model schematic diagram in an embodiment of the present application;

[0105] Figure 3 is an influence schematic diagram of gas flow shear force on liquid film fluctuation instability growth rate in an embodiment of the present application;

[0106] Figure 4 is an influence schematic diagram of evaporation effect on liquid film fluctuation instability growth rate in an embodiment of the present application. DETAILED DESCRIPTION

[0107] In order to make the purpose, technical scheme and advantages of the present application more clear, the following will further describe the disclosed embodiments of the present application in combination with the drawings.

[0108] According to the research status of liquid film flow, aiming at the deficiency that the current liquid film flow rarely considers the influences of gravity, gas flow shear force and evaporation effect, for small Reynolds number liquid film flow, a differential equation describing the spatial evolution of liquid film thickness with time is derived by using long fluctuation theory method, and a weak nonlinear evolution analysis method of fluctuation is used for analysis, and the influence law of gas flow shear force, evaporation effect, Reynolds number and Weber number on fluctuation instability growth rate is obtained.

[0109] Reference Figures 1-2 In the embodiment, the small Reynolds number liquid film flow simulation method comprises:

[0110] S1, dimensionless processing is performed on parameters to obtain dimensionless parameters.

[0111] In the embodiment, the dimensionless processing procedure of the parameters is as follows:

[0112] A small quantity ε = h0 / l is introduced. Wherein, h0 represents the initial liquid film thickness, and l represents a typical liquid-gas interface deformation wavelength.

[0113] A velocity of the liquid layer surface under no disturbance is introduced Wherein, ρ represents the density of the fluid in the liquid film, g represents the gravitational acceleration, and μ represents the viscosity at zero shear rate.

[0114] A Reynolds number Re = ρU0h0 / μ is introduced.

[0115] A Weber number We = γ0 / ρU02h0 is introduced. Wherein, γ0 represents the liquid film surface tension coefficient.

[0116] Based on ε, U0, Re and We, the liquid film flow direction coordinate x, the thickness direction coordinate y, the liquid film thickness h, the gas flow shear force τ g , the pressure p in the liquid film, and the external gas pressure p g, the velocity of the liquid film along the flow direction u, the normal velocity of the liquid film v, time t, the gas flow velocity U g and the liquid film surface velocity u s The dimensionless processing is carried out, and the dimensionless parameters are obtained:

[0117]

[0118] wherein, U gg and are the corresponding dimensionless parameters of x, y, h, σ xx , σ yy , σ xy , τ g , p, p g , u, v, t, U g and u s .

[0119] S2, for the liquid film flowing through the inclined rigid flat plate under the action of gravity, gas flow shear force and evaporation effect, the corresponding mass equation a and momentum equation a are given.

[0120] In this embodiment, the expression of the mass equation a is as follows:

[0121]

[0122] The expression of the momentum equation a is as follows:

[0123]

[0124] Wherein, θ represents the included angle between the flow direction of the liquid film and the horizontal direction.

[0125] S3, for the liquid film surface under the action of gravity, gas flow shear force, gas pressure and evaporation effect, the corresponding normal stress balance condition a, shear stress balance condition a, wall surface no-slip boundary condition a and liquid film surface motion boundary condition a under the influence of evaporation effect are given.

[0126] In this embodiment, the expression of the normal stress balance condition a is as follows:

[0127]

[0128] Wherein, J represents the mass jump flow of the liquid film due to the interface phase change; h t represents the change rate of the liquid film thickness with time t, h x represents the change rate of the liquid film thickness with x, h xx represents the second partial derivative of the liquid film thickness with respect to x, u xThis represents the partial derivative of the liquid film velocity along the flow direction with respect to x. u y This represents the partial derivative of the liquid film velocity along the flow direction with respect to y.

[0129] The expression for the shear stress equilibrium condition 'a' is as follows:

[0130]

[0131] The expression for the no-slip boundary condition 'a' on the wall is as follows:

[0132] u=0,v=0(y=0)···(5)

[0133] The expression for the boundary condition 'a' for the motion of the liquid film surface is as follows:

[0134]

[0135] Where h(x,t) represents the rate of change of liquid film thickness along the flow direction with time.

[0136] The preferred, known expression for J is as follows:

[0137]

[0138] ξ=1 / (Ku·Pr), Ku=ΔH / (C p ·ΔT), Pr=ν / a

[0139] Where k represents thermal conductivity, ΔH represents the latent heat of vaporization of the liquid, T represents the temperature inside the liquid film, Ku represents the Kutateladzes number, and C p ΔT represents the specific heat capacity of the fluid within the liquid film, ΔT represents the temperature change within the liquid film, Pr represents the Prandtl number, ν represents the kinematic viscosity coefficient, a represents the thermal diffusivity, and ξ represents the mass jump flow rate. A positive value of ξ indicates the condensation state, while a negative value indicates the evaporation state.

[0140] For the pneumatically heated surface of the material, we have:

[0141]

[0142] Therefore, the simplified expression for J is as follows:

[0143]

[0144] Where, q or This represents the heat flux density of the cold wall.

[0145] S4, based on the dimensionless parameters, the mass equation a, momentum equation a, normal stress balance condition a, shear stress balance condition a, wall no-slip boundary condition a and liquid film surface motion boundary condition a are respectively dimensionless, and the dimensionless mass equation b, momentum equation b, normal stress balance condition b, shear stress balance condition b, wall no-slip boundary condition b and liquid film surface motion boundary condition b are obtained.

[0146] In this embodiment, the expression of the mass equation b is as follows:

[0147]

[0148] The expression of the momentum equation b is as follows:

[0149]

[0150] The expression of the normal stress balance condition b is as follows:

[0151]

[0152] The expression of the shear stress balance condition b is as follows:

[0153]

[0154] The expression of the wall no-slip boundary condition b is as follows:

[0155]

[0156] The expression of the liquid film surface motion boundary condition b is as follows:

[0157]

[0158] wherein J A and J B are evaporation effect parameters,

[0159] S5, the Weber number We is scaled to make the scaled Weber number of the order of 1, the scaled Weber number is substituted into the normal stress balance condition b, and small quantities are omitted by order analysis to obtain the normal stress balance condition c.

[0160] In this embodiment, the Weber number We can be scaled in the following way:

[0161]

[0162] wherein, represents the scaled Weber number, and O(1) represents the order of 1.

[0163] Then, the expression of the normal stress balance condition c obtained by the order of magnitude analysis after neglecting small quantities is as follows:

[0164]

[0165] S6, the dimensionless liquid film velocity along the flow direction, the normal velocity of the liquid film and the pressure in the liquid film are respectively expanded by small quantity ε (ε << 1), and the zero-order and first-order quantities are reserved, which are substituted into the mass equation b, the momentum equation b, the normal stress balance condition c, the shear stress balance condition b, the wall no-slip boundary condition b and the liquid film surface movement boundary condition b, and simplified to obtain the zero-order solution and the first-order solution of the liquid film velocity along the flow direction.

[0166] In this embodiment, the solving process of the zero-order solution and the first-order solution of the liquid film velocity along the flow direction is as follows:

[0167] The dimensionless liquid film velocity along the flow direction, the normal velocity of the liquid film and the pressure in the liquid film are respectively expanded by small quantity ε:

[0168]

[0169] wherein, represents the dimensionless zero-order liquid film velocity along the flow direction, the normal velocity of the liquid film and the pressure in the liquid film, represents the dimensionless first-order liquid film velocity along the flow direction, the normal velocity of the liquid film and the pressure in the liquid film, O(ε 2 ) represents a small quantity of order ε 2 .

[0170] Substituting and into the mass equation b, the momentum equation b, the normal stress balance condition c, the shear stress balance condition b, the wall no-slip boundary condition b and the liquid film surface movement boundary condition b, the following simplification process is performed:

[0171]

[0172] wherein, represents the differential of , represents the differential of , represents the differential of , represents the differential of , represents the differential of , represents the differential of , represents the differential of , represents the differential of , represents the differential of The differential, express right The differential, express right The differential.

[0173] Based on equation (15), the zeroth-order solution of the velocity of the liquid film along the flow direction is obtained. and first-order solution

[0174]

[0175] in, express right The differential, express right Mixed differentials, express right The third differential.

[0176] S7. Integrate the mass equation b along the liquid film thickness direction, combine the wall no-slip boundary condition b and the liquid film surface motion boundary condition b, and substitute the zero-order and first-order solutions of the liquid film velocity along the flow direction obtained in step S6 to obtain the differential equation for the evolution of the liquid film thickness.

[0177] In this embodiment, the process of determining the differential equation is as follows:

[0178] Integrating mass equation b along the liquid film thickness direction, we get:

[0179]

[0180] q represents flow rate:

[0181]

[0182] Substituting the zero-order and first-order solutions of the liquid film velocity along the flow direction obtained in step S6 into formula (18), we get:

[0183]

[0184] Substituting equation (19) into equation (17), we obtain the differential equation for the evolution of the liquid film thickness:

[0185]

[0186] in, express right The fourth differential.

[0187] S8, based on the differential equation, wave weak nonlinear evolution analysis is carried out, and the variation law of disturbance instability growth rate with inclination angle, evaporation effect, airflow shear force, Reynolds number and Weber number is obtained.

[0188] In this embodiment, the wave weak nonlinear evolution analysis process is as follows:

[0189] Let Substitute the differential equation, ignore O(ε 2 ) to obtain:

[0190]

[0191] Wherein, η represents the liquid film thickness after substitution, represents the differential of η to , represents the differential of η to , represents the second differential of η to , represents the fourth differential of η to . Substitute: represents the dimensionless liquid film flow direction coordinate after substitution,

[0192] represents the dimensionless time after substitution, then:

[0193]

[0194] Wherein, represents the differential of η to , represents the differential of η to , represents the second differential of η to , represents the fourth differential of η to . Consider the flat plate solution, i.e. η≡0, then:

[0195]

[0196]

[0197] Given: Wherein, H<<1, i represents the imaginary unit, k represents the wave number, and ω represents the angular frequency.

[0198] Then:

[0199]

[0200] Then, the real part of ω is:

[0201] ​​​​​​

[0202] where ω r represents the real part of ω, which is used to characterize the growth rate of instability.

[0203] S9, the size of the evaporation effect and the gas flow shear force in step S8 is changed, a variation curve of the growth rate of instability is obtained, and the influence law of the evaporation effect and the gas flow shear force on the wave transmission instability growth rate of the liquid film is determined.

[0204] In the embodiment, it is given that U gg = 10.0, J A = J B = 1.0, the size of the gas flow shear force in the differential equation is changed, and the variation of the wave instability growth rate with k is obtained, as shown in Figure 3 Further, it is given that τ g = 0.0, J A = J B = J, the size of the evaporation effect in the differential equation is changed: 0.1, 0.5, 1.0, 2.0, and the variation of the wave instability growth rate with k is obtained, as shown in Figure 4

[0205] Although the present application has been disclosed with the above preferred embodiments, it is not intended to limit the present application, and any person skilled in the art can make possible changes and modifications to the technical solutions of the present application by using the disclosed methods and technical contents without departing from the spirit and scope of the present application. Therefore, any simple modification, equivalent change and modification made to the above embodiments according to the technical essence of the present application, which does not depart from the content of the technical solutions of the present application, belongs to the protection scope of the technical solutions of the present application.

[0206] The content not described in detail in the specification of the present application belongs to the known technology of the person skilled in the art.​

Claims

1. A method for simulating low Reynolds number liquid film flow, characterized in that, include: S1, the parameters are dimensionless to obtain dimensionless parameters; S2, for the flow of liquid film over an inclined rigid plate under gravity, airflow shear force and evaporation effect, the corresponding mass equation a and momentum equation a are given; S3 provides the corresponding normal stress equilibrium condition a, shear stress equilibrium condition a, wall no-slip boundary condition a, and liquid film surface motion boundary condition a under the influence of evaporation effect for the liquid film surface under the action of gravity, airflow shear force, gas pressure and evaporation effect. S4. Based on dimensionless parameters, the mass equation a, momentum equation a, normal stress equilibrium condition a, shear stress equilibrium condition a, wall no-slip boundary condition a, and liquid film surface motion boundary condition a are respectively dimensionless to obtain dimensionless mass equation b, momentum equation b, normal stress equilibrium condition b, shear stress equilibrium condition b, wall no-slip boundary condition b, and liquid film surface motion boundary condition b. S5. Scale the Weber number We so that the scaled Weber number is on the order of 1. Substitute the scaled Weber number into the normal stress equilibrium condition b, and use order of magnitude analysis to neglect small quantities to obtain the normal stress equilibrium condition c. S6. Expand the dimensionless liquid film velocity along the flow direction, the normal velocity of the liquid film, and the pressure inside the liquid film using small quantities ε, retaining the zero-order and first-order quantities. Substitute these quantities into the mass equation b, momentum equation b, normal stress equilibrium condition c, shear stress equilibrium condition b, wall no-slip boundary condition b, and liquid film surface motion boundary condition b, and simplify to obtain the zero-order and first-order solutions for the liquid film velocity along the flow direction; where ε << 1. S7. Integrate the mass equation b along the liquid film thickness direction, combine the wall no-slip boundary condition b and the liquid film surface motion boundary condition b, and substitute the zero-order and first-order solutions of the liquid film velocity along the flow direction obtained in step S6 to obtain the differential equation for the evolution of the liquid film thickness. S8, based on differential equations, performs wave weak nonlinear evolution analysis to obtain the variation law of the growth rate of disturbance instability with tilt angle, evaporation effect, airflow shear force, Reynolds number and Weber number; S9. Change the magnitude of the evaporation effect and the airflow shear force in step S8 to obtain the change curve of the instability growth rate, and determine the influence of the evaporation effect and the airflow shear force on the growth rate of liquid film wave transmission instability.

2. The method for simulating low Reynolds number liquid film flow according to claim 1, characterized in that, The parameters are dimensionless to obtain dimensionless parameters, including: Introduce a small amount ε = h0 / l; where h0 represents the initial liquid film thickness and l represents the typical deformation wavelength of the liquid-gas interface; Introducing velocity to the undisturbed lower liquid layer surface Where ρ represents the density of the fluid in the liquid film, g represents the gravitational acceleration, and μ represents the viscosity at zero shear rate; Introduce the Reynolds number Re = ρU0h0 / μ; Introducing the Weber number Wherein, γ0 represents the surface tension coefficient of the liquid film; Based on ε, U0, Re, and We, the flow direction coordinate x, thickness direction coordinate y, liquid film thickness h, and airflow shear force τ are considered. g Liquid film internal pressure p, external gas pressure p g The velocity of the liquid film along the flow direction (u), the normal velocity of the liquid film (v), time (t), and the airflow velocity (U) g and liquid film surface velocity u s Dimensionalization is performed to obtain dimensionless parameters: in, U gg and They are x, y, h, and σ, respectively. xx σ yy σ xy τ g p, p g u, v, t, U g and u s The corresponding dimensionless parameter.

3. The method for simulating low Reynolds number liquid film flow according to claim 2, characterized in that, The expression for mass equation a is as follows: The expression for the momentum equation a is as follows: Where θ represents the angle between the direction of liquid film flow and the horizontal direction; The expression for the normal stress equilibrium condition 'a' is as follows: Where J represents the mass jump flow rate of the liquid film caused by the interfacial phase transition; h t This represents the rate of change of the liquid film thickness with time t. h x This represents the rate of change of the liquid film thickness with respect to x. h xx This represents the second partial derivative of the liquid film thickness with respect to x. u x This represents the partial derivative of the liquid film velocity along the flow direction with respect to x. u y This represents the partial derivative of the liquid film velocity along the flow direction with respect to y. The expression for the shear stress equilibrium condition 'a' is as follows: The expression for the no-slip boundary condition 'a' on the wall is as follows: u=0,v=0(y=0)···(5) The expression for the boundary condition 'a' for the motion of the liquid film surface is as follows: Where h(x,t) represents the rate of change of liquid film thickness along the flow direction with time.

4. The method for simulating low Reynolds number liquid film flow according to claim 3, characterized in that, The original expression for J is as follows: ξ=1 / (Ku·Pr),Ku=ΔH / (C p ·ΔT),Pr=ν / a Where k represents thermal conductivity, ΔH represents the latent heat of vaporization of the liquid, T represents the temperature inside the liquid film, Ku represents the Kutateladzes number, and C p Let denot ΔT represent the specific heat capacity of the fluid within the liquid film, ΔT represent the temperature change within the liquid film, Pr represent the Prandtl number, ν represent the kinematic viscosity coefficient, and a represent the thermal diffusivity; ξ represents the mass jump flow rate, with a positive value indicating condensation and a negative value indicating evaporation; for a pneumatically heated surface of the material, we have: Therefore, the simplified expression for J is as follows: Where, q or This represents the heat flux density of the cold wall.

5. The method for simulating low Reynolds number liquid film flow according to claim 4, characterized in that, The expression for mass equation b is as follows: The expression for the momentum equation b is as follows: The expression for the normal stress equilibrium condition b is as follows: The expression for the shear stress equilibrium condition b is as follows: The expression for the no-slip boundary condition b on the wall is as follows: The expression for the boundary condition b for the liquid film surface motion is as follows: Among them, J A and J B For evaporation effect parameters, 6. The method for simulating low Reynolds number liquid film flow according to claim 5, characterized in that, The Weber number We is scaled as follows: in, This represents the scaled Weber number, and O(1) indicates that the order of magnitude is 1.

7. The method for simulating low Reynolds number liquid film flow according to claim 6, characterized in that, The expression for the normal stress equilibrium condition c is as follows:

8. The method for simulating low Reynolds number liquid film flow according to claim 7, characterized in that, Expand the dimensionless liquid film velocity along the flow direction, the normal velocity of the liquid film, and the pressure inside the liquid film using small quantities ε. Retaining the zeroth and first-order quantities, substitute them into the mass equation b, momentum equation b, normal stress equilibrium condition c, shear stress equilibrium condition b, no-slip boundary condition b, and liquid film surface motion boundary condition b. After simplification, obtain the zeroth and first-order solutions for the liquid film velocity along the flow direction, including: Expand the dimensionless liquid film velocity along the flow direction, the normal velocity of the liquid film, and the pressure inside the liquid film using a small quantity ε: in, This represents the velocity along the flow direction, the normal velocity of the liquid film, and the pressure within the liquid film in a dimensionless zero-order liquid film. O(ε) represents the dimensionless velocity along the flow direction of the first-order liquid film, the normal velocity of the liquid film, and the pressure within the liquid film. 2 ) indicates a magnitude of ε 2 Small quantities; Will and Substituting the mass equation b, momentum equation b, normal stress equilibrium condition c, shear stress equilibrium condition b, no-slip boundary condition b, and liquid film surface motion boundary condition b, the following simplification is performed: in, express right The differential, express right The differential, express right The differential, express right The differential, express right The differential, express right The differential, express right The differential, express right The differential; Based on equation (15), the zeroth-order solution of the velocity of the liquid film along the flow direction is obtained. and first-order solution in, express right The differential, express right Mixed differentials, express right The third differential.

9. The method for simulating low Reynolds number liquid film flow according to claim 8, characterized in that, Integrating mass equation b along the liquid film thickness direction, combining the no-slip boundary condition b on the wall and the motion boundary condition b on the liquid film surface, and substituting the zero-order and first-order solutions of the liquid film velocity along the flow direction obtained in step S6, we obtain the differential equation for the evolution of the liquid film thickness, including: Integrating mass equation b along the liquid film thickness direction, we get: q represents flow rate: Substituting the zero-order and first-order solutions of the liquid film velocity along the flow direction obtained in step S6 into formula (18), we get: Substituting equation (19) into equation (17), we obtain the differential equation for the evolution of the liquid film thickness: in, express right The fourth differential.

10. The method for simulating low Reynolds number liquid film flow according to claim 9, characterized in that, Based on differential equations, weakly nonlinear evolution analysis of the waves was conducted to obtain the variation of the growth rate of disturbance instability with tilt angle, evaporation effect, airflow shear force, Reynolds number, and Weber number, including: make Substituting into the differential equation, neglecting O(ε) 2 )get: Where η represents the thickness of the liquid film after the substitution. Indicates η pairs The differential, Indicates η pairs The differential, Indicates η pairs The second differential, Indicates η pairs The fourth differential; Substitution: Represents the coordinates of the dimensionless liquid film flow direction after the substitution. Let the dimensionless time after the substitution be represented, then: in, Indicates η pairs The differential, Indicates η pairs The differential, Indicates η pairs The second differential, Indicates η pairs The fourth differential; Considering the solution for the flat plate, i.e., η≡0, we have: Given: Where H << 1, i represents the imaginary unit, k represents the wave number, and ω represents the angular frequency; Then we have: Then, the real part of ω is: Where, ω r ω represents the real part of ω, used to characterize the rate of instability growth.

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