Wall heat flux prediction method based on integral form

Abstract

The present invention discloses a wall heat flux prediction method based on an integral form. Based on the Reynolds-averaged energy equation of the fluid, an integral method is used to establish a quantitative relationship between the wall heat flux of the turbulent boundary layer and the turbulent statistics; the wall normal gradient of the total heat flux density is used to replace the streamwise gradient term in the quantitative relationship, and the wall heat flux equation containing the total heat flux density is obtained by changing the upper and lower limits of the integral; a near-wall total heat flux density model is established and used to replace the total heat flux density in the wall heat flux equation to obtain a wall heat flux prediction model in a zero-pressure gradient turbulent boundary layer; the average temperature and normal heat flux density data of the fluid at the previous position in the flow direction are collected, and the wall heat flux is obtained according to the proposed wall heat flux prediction model. The method disclosed by the present invention has the characteristics of high accuracy, low computational complexity, and independence from near-wall data. It can accurately predict the wall heat flux using the flow field information of the outer layer of the boundary layer, and the prediction result is insensitive to changes in the thickness of the boundary layer.

Description

Technical Field

[0001] The invention relates to a wall heat flow prediction method based on integral form, and belongs to the field of thermal measurement control. Background Art

[0002] Turbulent boundary layer heat transfer is a widespread phenomenon in nature, industry, and the environment, for example in energy exchange via ocean convection, nuclear reactor core cooling, and waste heat recovery from flue gas in thermal power plants. This phenomenon is often accompanied by severe heat dissipation, which in turn increases energy consumption. Wall heat flux, reflecting the heat transfer power near the wall, is an important parameter studied in turbulent boundary layer heat transfer. Therefore, accurately predicting and controlling wall heat flux in turbulent boundary layers is crucial for reducing energy consumption.

[0003] Wall heat flux measurement methods can be categorized as direct or indirect. Common direct methods include thermal and optical measurement. Thermal measurement uses thermocouples to measure the temperature gradient near the wall. However, due to interference between the thermocouple probe and the wall, the temperature gradient measurement error is large. The measurement accuracy of optical measurement methods is easily affected by fluid transparency and background light noise. Indirect methods primarily include analogy and inversion. The choice of analog form in the analogy method significantly influences the accuracy of the measurement results, while the application of the inversion method has certain limitations.

[0004] In addition to experimental methods, some researchers use turbulent boundary layer equations to predict wall heat flux. Such methods generally rely on near-wall data. The accuracy of the experimentally determined temperature near the wall is limited by the spatial resolution of the measurement system. This limitation results in large errors in near-wall temperature measurements. Therefore, accurately predicting wall heat flux remains challenging.

[0005] Therefore, it is necessary to conduct more in-depth research on wall heat flux prediction methods and develop some high-precision methods to predict wall heat flux in order to solve the above problems. Summary of the Invention

[0006] In order to overcome the above problems, the inventors conducted in-depth research and proposed a wall heat flux prediction method based on integral form, which includes the following steps:

[0007] S1. Establish a quantitative relationship between the wall heat flux of the turbulent boundary layer and the turbulence statistics;

[0008] S2. Replace the streamwise gradient term in the quantitative relationship with the wall normal gradient of the total heat flux density. By changing the upper and lower limits of the integral and ignoring the difficult-to-obtain near-wall data, the wall heat flux equation containing the total heat flux density is obtained.

[0009] S3. According to the flow characteristics of the turbulent boundary layer near the wall, a total heat flux density model near the wall is established;

[0010] S4. Substituting the total heat flux density in the wall heat flux equation with the near-wall total heat flux model, a wall heat flux prediction model in the zero pressure gradient turbulent boundary layer is obtained;

[0011] S5. Collect the average temperature and normal heat flux density data of the fluid at the previous position in the flow direction, and obtain the wall heat flux based on the wall heat flux prediction model in the zero pressure gradient turbulent boundary layer.

[0012] In a preferred embodiment, in S1, based on the Reynolds-averaged energy equation of the fluid, an integral method is used to establish a quantitative relationship between the heat flux on the wall of the turbulent boundary layer and the turbulence statistics.

[0013] In a preferred embodiment, in S1, the Reynolds average energy equation is:

[0014]

[0015] in, is the convection term, y is the wall normal coordinate, Re δ is the boundary layer thickness Reynolds number, Pr is the Prandtl number, is the dimensionless mean temperature, is the normal heat flux.

[0016] In a preferred embodiment, the Reynolds-averaged energy equation is integrated three times along the normal direction to obtain a quantitative relationship between the wall heat flux and the turbulence statistics.

[0017] In a preferred embodiment, the quantitative relationship between the turbulent boundary layer wall heat flux and the turbulence statistics is expressed as:

[0018]

[0019] Where β is the distance from the upper limit of the integral to the wall, is the total heat flux, is the Stanton number, is the wall heat flux.

[0020] In a preferred embodiment, in S2, based on the quantitative relationship between the wall heat flux of the turbulent boundary layer and the turbulence statistics, the wall heat flux equation containing the total heat flux density is obtained by integrating from α / δ to β / δ along the wall normal, where δ is the boundary layer thickness.

[0021] In a preferred embodiment, in S3, the total heat flux density model near the wall is expressed as:

[0022]

[0023] Among them, q +is the dimensionless total heat flux, δ is the boundary layer thickness, and a is a configurable parameter.

[0024] In a preferred embodiment, in S4, the wall heat flux prediction model in the zero pressure gradient turbulent boundary layer is expressed as:

[0025]

[0026] Among them, C3 is the coefficient.

[0027] The present invention also provides an electronic device, comprising:

[0028] At least one processor; and a memory communicatively connected to the at least one processor; wherein the memory stores instructions executable by the at least one processor, and the instructions are executed by the at least one processor to enable the at least one processor to perform any one of the above methods.

[0029] The present invention also provides a computer-readable storage medium storing computer instructions, wherein the computer instructions are used to enable the computer to execute any one of the above methods.

[0030] The beneficial effects of the present invention include:

[0031] (1) Significantly reduced dependence on near-wall data;

[0032] (2) It has the characteristics of high precision and small computational complexity. It can use the flow field information of the outer layer of the boundary layer (average temperature and normal heat flux density) to accurately predict the wall heat flux, with a prediction error within 4%.

[0033] (3) The prediction results are insensitive to changes in boundary layer thickness. BRIEF DESCRIPTION OF THE DRAWINGS

[0034] Figure 1 2. It is a flow chart of a wall heat flux prediction method based on integral form according to a preferred embodiment of the present invention;

[0035] Figure 2 is the error between the result obtained in Example 1 and the wall heat flux result recorded in the corresponding literature;

[0036] Figure 3 is the error between the result obtained in Example 2 and the wall heat flux result recorded in the corresponding literature;

[0037] Figure 4 3 is a comparison diagram of the wall heat flux results obtained in Example 1 and Comparative Example 1, wherein (a) is the result obtained in Comparative Example 1, and (b) is the result obtained in Example 1. DETAILED DESCRIPTION

[0038] The present invention will be described in further detail below with reference to the accompanying drawings and examples, through which the features and advantages of the present invention will become more clearly understood.

[0039] The word "exemplary" is used exclusively herein to mean "serving as an example, example, or illustration." Any embodiment described herein as "exemplary" is not necessarily to be construed as preferred or advantageous over other embodiments. Although various aspects of the embodiments are shown in the drawings, the drawings are not necessarily drawn to scale unless otherwise indicated.

[0040] According to the present invention, a wall heat flux prediction method based on integral form is provided, such as Figure 1 As shown, the following steps are included:

[0041] S1. Establish a quantitative relationship between the wall heat flux of the turbulent boundary layer and the turbulence statistics;

[0042] S2. Replace the streamwise gradient term in the quantitative relationship with the wall normal gradient of the total heat flux density. By changing the upper and lower limits of the integral and ignoring the difficult-to-obtain near-wall data, the wall heat flux equation containing the total heat flux density is obtained.

[0043] S3. According to the flow characteristics of the turbulent boundary layer near the wall, a total heat flux density model near the wall is established;

[0044] S4. Substituting the total heat flux density in the wall heat flux equation with the near-wall total heat flux model, a wall heat flux prediction model in the zero pressure gradient turbulent boundary layer is obtained;

[0045] S5. Collect the average temperature and normal heat flux density data of the fluid at the previous position in the flow direction, and obtain the wall heat flux based on the wall heat flux prediction model in the zero pressure gradient turbulent boundary layer.

[0046] In a preferred embodiment, in S1, based on the Reynolds-averaged energy equation of the fluid, an integral method is used to establish a quantitative relationship between the heat flux on the wall of the turbulent boundary layer and the turbulence statistics.

[0047] Further preferably, the Reynolds averaged energy equation is integrated three times along the normal direction to obtain a quantitative relationship between the wall heat flux and the turbulence statistics.

[0048] According to the present invention, in S1, the Reynolds average energy equation is:

[0049]

[0050] in, is the convection term, y is the wall normal coordinate, Re δ is the boundary layer thickness Reynolds number, Pr is the Prandtl number, is the dimensionless mean temperature, is the normal heat flux.

[0051] Among them, the convection term Expressed as:

[0052]

[0053] Where x is the flow direction coordinate, is the average velocity of the fluid flow, is the average normal velocity of the fluid, is the flow heat flux density;

[0054] The dimensionless temperature is expressed as:

[0055] Θ=(TT w ) / (T ∞ -T w ),

[0056] T is the flow field temperature, T w is the wall temperature, T ∞ is the free flow temperature;

[0057] The boundary layer thickness Reynolds number is expressed as:

[0058] Re δ =U ∞ δ / ν

[0059] Among them, U ∞ is the free stream velocity, δ is the boundary layer thickness, and ν is the fluid kinematic viscosity;

[0060] The Prandtl number is expressed as:

[0061] Pr=ρνc p / k

[0062] Where ρ is the fluid density, c p is the constant pressure specific heat of the fluid, and k is the thermal conductivity of the fluid.

[0063] According to the present invention, the quantitative relationship between the wall heat flux of the turbulent boundary layer and the turbulence statistics is expressed as:

[0064]

[0065] Where β is the distance from the upper limit of the integral to the wall, q is the total heat flux, St is the Stanton number, q w is the wall heat flux.

[0066] Further, according to the present invention, the total heat flux The Stanton number is The wall heat flux is

[0067] In S2, the wall normal gradient of the total heat flux density is used to replace the streamwise gradient term in the quantitative relationship. In this way, the wall heat flux can be obtained by measuring the experimental data at only one position in the streamwise direction.

[0068] Preferably, in S2, by changing the upper and lower limits of the integral and ignoring the difficult-to-obtain near-wall data, the dependence of the wall heat flux prediction on the near-wall data is significantly reduced, thereby solving the problem of inaccurate wall heat flux prediction caused by the difficulty in collecting near-wall data.

[0069] Specifically, based on the quantitative relationship between the wall heat flux and turbulent statistics in the turbulent boundary layer, the wall heat flux equation containing the total heat flux density is obtained by integrating from α / δ to β / δ along the wall normal, which is expressed as:

[0070]

[0071] Where β is the distance from the upper limit of the integral to the wall, and α is the distance from the lower limit of the integral to the wall.

[0072] In S3, the total heat flux density model near the wall is expressed as:

[0073]

[0074] Among them, q + is the dimensionless total heat flux, δ is the boundary layer thickness, and a is a configurable parameter.

[0075] Preferably, q + =q / q w , q is the total heat flux, q w is the wall heat flux.

[0076] Preferably, the parameter can be obtained through simulation fitting and set to a=-1.36.

[0077] In S4, the wall heat flux prediction model in the zero pressure gradient turbulent boundary layer is expressed as:

[0078]

[0079] Among them, C3 is the coefficient.

[0080] In a preferred embodiment, the coefficient C3 is expressed as:

[0081]

[0082] In a preferred embodiment, the lower limit of integration is set to α + =u τ α / ν, where u τ is the viscous velocity scale, α + Refers to the dimensionless lower limit of integration, preferably, α+ =100; the upper limit of the integral is β / δ=0.4.

[0083] The wall heat flux prediction model obtained by the present invention can accurately predict the wall heat flux by using the flow field information of the outer layer of the boundary layer. The error of the final wall heat flux prediction value is within 4%, which significantly reduces the dependence of the wall heat flux prediction on near-wall data. The prediction result is insensitive to changes in the boundary layer thickness.

[0084] Various embodiments of the methods described above in the present invention may be implemented in digital electronic circuit systems, integrated circuit systems, field programmable gate arrays (FPGAs), application specific integrated circuits (ASICs), application specific standard products (ASSPs), system-on-chip systems (SOCs), programmable logic devices (CPLDs), computer hardware, firmware, software, and / or combinations thereof. These various embodiments may include being implemented in one or more computer programs that are executable and / or interpreted on a programmable system comprising at least one programmable processor, which may be a special-purpose or general-purpose programmable processor that can receive data and instructions from a storage system, at least one input device, and at least one output device, and transmit data and instructions to the storage system, the at least one input device, and the at least one output device.

[0085] It should be understood that the various forms of the processes shown above can be used to reorder, add, or delete steps. For example, the steps described in this disclosure can be performed in parallel, sequentially, or in a different order, as long as the desired results of the technical solutions disclosed in this disclosure can be achieved. This is not a limitation herein.

[0086] Example

[0087] Example 1

[0088] An experiment was conducted to predict the wall heat flux of a turbulent boundary layer, which included the following steps:

[0089] S1. Establish a quantitative relationship between the wall heat flux of the turbulent boundary layer and the turbulence statistics;

[0090] S2. Replace the streamwise gradient term in the quantitative relationship with the wall normal gradient of the total heat flux density. By changing the upper and lower limits of the integral and ignoring the difficult-to-obtain near-wall data, the wall heat flux equation containing the total heat flux density is obtained.

[0091] S3. According to the flow characteristics of the turbulent boundary layer near the wall, a total heat flux density model near the wall is established;

[0092] S4. Using the near-wall total heat flux model to replace the total heat flux in the wall heat flux equation, a wall heat flux prediction model in the zero-pressure gradient turbulent boundary layer is obtained;

[0093] S5. Collect the average temperature and normal heat flux density data of the fluid at the previous position in the flow direction, and obtain the wall heat flux based on the wall heat flux prediction model in the zero pressure gradient turbulent boundary layer.

[0094] In S1, based on the Reynolds-averaged energy equation of the fluid, the quantitative relationship between the wall heat flux and the turbulence statistics is obtained by the integration method and is expressed as:

[0095]

[0096] In S2, the wall heat flux equation containing the total heat flux density is obtained, which is expressed as:

[0097]

[0098] In S3, the total heat flux density model near the wall is expressed as:

[0099]

[0100] Among them, q + =q / q w , a=-1.36.

[0101] In S4, the wall heat flux prediction model in the zero pressure gradient turbulent boundary layer is expressed as:

[0102]

[0103]

[0104] Among them, the lower limit of the integral is α + =100, and the upper limit of the integral is β / δ=0.4.

[0105] In S5, the wall heat flux is obtained based on the wall heat flux prediction model using the data published in existing literature. The literature involved is:

[0106] Document 1, Li D, Luo K, Fan J. Direct numerical simulation of heat transfer in a spatially developing turbulent boundary layer [J]. Physics of Fluids, 2016, 28(10);

[0107] Reference 2: Wu X, Moin P, Wallace JM, et al. Transitional-turbulent spots and turbulent-turbulent spots in boundary layers[J]. Proceedings of the National Academy of Sciences, 2017, 114(27): E5292-E5299;

[0108] Reference 3: Araya G, Castillo L. DNS of turbulent thermal boundary layers up to Reθ=2300[J]. International journal of heat and mass transfer, 2012, 55(15-16): 4003-4019;

[0109] Reference 4: Araya G, Castillo L. Direct numerical simulations of turbulent thermal boundary layers subjected to adverse streamwise pressure gradients[J]. Physics of Fluids, 2013, 25(9);

[0110] Reference 5: Balasubramanian AG, Guastoni L, Schlatter P, et al. Direct numerical simulation of a zero-pressure-gradient turbulent boundary layer with passive scalars up to Prandtl number Pr=6[J]. Journal of Fluid Mechanics, 2023, 974:A49;

[0111] Document 6. Houra T, Nagano Y. Effects of adverse pressure gradient on heattransfer mechanism in thermal boundary layer [J]. International journal of heat and fluid flow, 2006, 27(5):967-976;

[0112] Document 7. Nagata K, Sakai Y, Komori S. Effects of small-scale freestream turbulence on turbulent boundary layers with and without thermal convection[J]. Physics of Fluids, 2011, 23(6).

[0113] Example 2

[0114] The same experiment as in Example 1 was conducted, except that in S4, the boundary layer thickness δ was varied by ±25%; in S5, the wall heat flux was obtained according to the wall heat flux prediction model using direct numerical simulation data from the following literatures:

[0115] Document 8, Wu,

[0116] Document 9, Li D, Luo K, Fan J. Direct numerical simulation of heat transfer in a spatially developing turbulent boundary layer [J]. Physics of Fluids, 2016, 28(10);

[0117] Document 10. Balasubramanian AG, Guastoni L, Schlatter P, et al. Directnumerical simulation of a zero-pressure-gradient turbulent boundary layer with passive scalars up to Prandtl number Pr=6[J]. Journal of FluidMechanics, 2023,974:A49.

[0118] Comparative Example 1

[0119] The same experiment as in Example 1 was performed, except that the wall heat flux was obtained using the wall heat flux prediction method proposed by Mehdi.

[0120] The detailed process of Mehdi's wall heat flux prediction method can be found in the reference Ebadi A, Mehdi F, White C M. An exact integral method to evaluate wall heat flux in spatially developing two-dimensional wall-bounded flows[J]. International Journal of Heat and Mass Transfer, 2015, 84: 856-861.

[0121] The literature involved in the data used is:

[0122] Document 11, Li D, Luo K, Fan J. Direct numerical simulation of heattransfer in a spatially developing turbulent boundary layer [J]. Physics of Fluids, 2016, 28(10).

[0123] References 1 to 7 record the direct numerical simulation data and experimental data of the wall heat flux of the zero pressure gradient turbulent boundary layer. The wall heat flux obtained in Example 1 is denoted as St f,p , the wall heat flux recorded in the literature is recorded as St f,t , the wall heat flux prediction error is defined as The results are as follows Figure 2 As shown, the horizontal axis is the momentum thickness Reynolds number, U ∞is the free stream velocity, θ is the momentum thickness of the flat plate boundary layer, ν is the fluid kinematic viscosity, and the ordinate is the wall heat flux prediction error. As can be seen from the figure, the difference between the results obtained in Example 1 and the results obtained in the literature is less than 4%, indicating that the prediction error in Example 1 is within 4%;

[0124] References 8 to 10 record the direct numerical simulation data of the wall heat flux of the zero pressure gradient turbulent boundary layer. The wall heat flux obtained in Example 2 is recorded as St f,p , the wall heat flux recorded in the literature is recorded as St f,t , the wall heat flux prediction error is defined as The results are as follows Figure 3 As shown, where the horizontal axis δ / δ t is the ratio of the nominal boundary layer thickness to the standard value, and the ordinate is the wall heat flux prediction error. As can be seen from the figure, when the boundary layer thickness δ varies by ±25%, the error of the results obtained in Example 2 remains within 4%, indicating that the wall heat flux prediction is insensitive to changes in boundary layer thickness.

[0125] The direct numerical simulation results of the zero pressure gradient turbulent boundary layer wall heat flux recorded in Reference 11 are taken as the true value, and the wall heat flux obtained in Example 1 and Comparative Example 1 are compared. The wall heat flux obtained in Example 1 and Comparative Example 1 is recorded as St f,p , the wall heat flux recorded in the literature is recorded as St f,t , the wall heat flux error is defined as The results are as follows Figure 4 As shown, Figure 4 (a) is the wall heat flux error obtained in Comparative Example 1, Figure 4 (b) shows the wall heat flux error obtained in Example 1. The abscissa represents the lower limit of integration, and the ordinate represents the upper limit of integration. As can be seen from the figure, the wall heat flux error obtained by the method in Example 1 is smaller than the wall heat flux error obtained by the method proposed by Mehdi in Comparative Example 1.

[0126] The present invention has been described above with reference to preferred embodiments, but these embodiments are merely exemplary and serve only as illustrations. On this basis, various replacements and improvements can be made to the present invention, all of which fall within the scope of protection of the present invention.

Claims

1. A wall heat flux prediction method based on integral form, characterized in that: The following steps are involved: S1. Establish a quantitative relationship between the wall heat flux of the turbulent boundary layer and the turbulence statistics; S2. Replace the streamwise gradient term in the quantitative relationship with the wall normal gradient of the total heat flux density. By changing the upper and lower limits of the integral and ignoring the difficult-to-obtain near-wall data, the wall heat flux equation containing the total heat flux density is obtained. S3. According to the flow characteristics of the turbulent boundary layer near the wall, a total heat flux density model near the wall is established; S4. Substituting the total heat flux density in the wall heat flux equation with the near-wall total heat flux model, a wall heat flux prediction model in the zero pressure gradient turbulent boundary layer is obtained; S5. Collecting the average temperature and normal heat flux density data of the fluid at the previous position in the upstream direction, and obtaining the wall heat flux according to the wall heat flux prediction model in the zero pressure gradient turbulent boundary layer; The quantitative relationship between the wall heat flux of the turbulent boundary layer and the turbulence statistics is expressed as: Where β is the distance from the upper limit of the integral to the wall, is the total heat flux, is the Stanton number, is the wall heat flux, y is the wall normal coordinate, Re δ is the boundary layer thickness Reynolds number, Pr is the Prandtl number, is the dimensionless mean temperature, is the normal heat flux density, ρ is the fluid density, c p is the constant pressure specific heat of the fluid, k is the thermal conductivity of the fluid, T w is the wall temperature, T ∞ is the free flow temperature, U ∞ is the free stream velocity; In S2, based on the quantitative relationship between the wall heat flux of the turbulent boundary layer and the turbulent statistics, the wall heat flux equation containing the total heat flux density is obtained by integrating from αδ to βδ along the wall normal, where δ is the boundary layer thickness and α is the distance from the lower limit of integration to the wall. In S3, the total heat flux density model near the wall is expressed as: Among them, q + is the dimensionless total heat flux, δ is the boundary layer thickness, and a is a configurable parameter.

2. The wall heat flux prediction method based on integral form according to claim 1 is characterized in that: In S1, based on the Reynolds-averaged energy equation of the fluid, an integral method is used to establish a quantitative relationship between the wall heat flux of the turbulent boundary layer and the turbulent statistics.

3. The wall heat flux prediction method based on integral form according to claim 2 is characterized in that: In S1, the Reynolds average energy equation is: in, is the convection term.

4. The wall heat flux prediction method based on integral form according to claim 3 is characterized in that: The Reynolds-averaged energy equation is integrated three times along the normal direction, and a quantitative relationship between the wall heat flux and the turbulence statistics is obtained.

5. The wall heat flux prediction method based on integral form according to claim 1 is characterized in that: In S4, the wall heat flux prediction model in the zero pressure gradient turbulent boundary layer is expressed as: Among them, C3 is the coefficient.

6. An electronic device comprising: at least one processor; and a memory communicatively connected to the at least one processor; wherein the memory stores instructions executable by the at least one processor, and the instructions are executed by the at least one processor to enable the at least one processor to execute the method according to any one of claims 1 to 5.

7. A computer-readable storage medium storing computer instructions, wherein: The computer instructions are used to cause the computer to execute the method according to any one of claims 1 to 5.

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