Momentum integral form of wall friction prediction method

Abstract

The present invention discloses a method for predicting wall friction resistance in the form of momentum integral, comprising the following steps: based on the average momentum equation of the fluid, an integral method is used to establish a quantitative relationship between the wall friction resistance of the turbulent boundary layer and turbulent statistics; by changing the upper and lower limits of the integral, difficult-to-obtain near-wall data is ignored to obtain a wall friction resistance equation containing total shear stress; based on the flow characteristics near the wall of the turbulent boundary layer, a near-wall total shear stress model is proposed; and this model is used to replace the total shear stress in the wall friction resistance equation to obtain a wall friction resistance prediction model in a zero-pressure gradient turbulent boundary layer. The wall friction resistance prediction method disclosed by the present invention has the characteristics of high accuracy, low computational complexity, and strong versatility, and can accurately predict wall friction resistance using the flow field information of the outer layer of the boundary layer.

Description

Technical Field

[0001] The invention relates to a method for predicting wall friction resistance in the form of momentum integration, and belongs to the field of force measurement and control. Background Art

[0002] The wall friction resistance in the turbulent boundary layer has a direct impact on energy consumption in the industrial field. The wall friction resistance in the turbulent boundary layer accounts for a large proportion of the total resistance. Reducing the wall friction resistance can not only save energy, but also increase the speed of transportation vehicles such as aircraft and ships.

[0003] Methods for measuring wall friction resistance can be divided into direct measurement and indirect measurement methods. Direct measurement methods use the displacement or deformation of stress-sensitive units such as floats for measurement, such as the microbalance method. It has the advantages of easy deployment, reasonable price, and little interference with the flow field. However, the microbalance is easily contaminated by attachments and difficult to clean. Indirect measurement methods are more common measurement methods. They use the principle of near-wall flow similarity for measurement, such as the near-wall velocity distribution method, the near-wall pressure measurement method, and the oil film interferometry method. Although these indirect measurement methods can measure wall friction resistance relatively accurately, each still has certain limitations. For example, there is a problem with probe positioning in near-wall velocity measurement, and the oil film interferometry method is limited in that the oil film thickness changes slowly under low shear stress conditions, making effective measurement impossible. Therefore, there is currently no reliable commercial instrument that can clearly define the measurement error to measure wall friction resistance.

[0004] In addition to experimental measurements, some researchers have also used the logarithmic law of turbulent boundary layers and boundary layer equations to estimate wall friction. However, these methods require data near the wall and at multiple flow locations. Due to the limited resolution of experimental instruments near the wall, measuring near-wall flow information is difficult. Consequently, these methods exhibit significant errors and are difficult to meet practical engineering requirements.

[0005] Therefore, it is necessary to conduct more in-depth research on the measurement method of wall friction resistance 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 method for predicting wall friction resistance in the form of momentum integration, which includes the following steps:

[0007] S1. Based on the mean momentum equation of the fluid, the integration method is used to establish the quantitative relationship between the wall friction resistance of the turbulent boundary layer and the turbulence statistics;

[0008] S2. By changing the upper and lower limits of integration and ignoring the difficult-to-obtain near-wall data, the wall friction resistance equation containing the total shear stress is obtained;

[0009] S3. Based on the flow characteristics of the turbulent boundary layer near the wall, a total shear stress model near the wall is proposed;

[0010] S4. Use the near-wall shear stress model to replace the total shear stress in the wall friction drag equation to obtain a wall friction drag prediction model in a zero-pressure-gradient turbulent boundary layer.

[0011] S5. Collect the average velocity and Reynolds stress data of the fluid at the previous position, and obtain the wall friction resistance according to the wall friction resistance prediction model.

[0012] In a preferred embodiment, in S1, the average momentum equation is:

[0013]

[0014] Among them, the convection term x is the streamwise coordinate, y is the wall normal coordinate, is the average velocity of the fluid flow, is the average normal velocity of the fluid, and is the Reynolds stress, and ν is the fluid kinematic viscosity.

[0015] In a preferred embodiment, in S1, the average momentum equation is integrated three times along the normal direction to obtain the quantitative relationship between the wall friction resistance and the turbulence statistics, which is expressed as:

[0016]

[0017] Among them, C f is the wall friction resistance, is the Reynolds number of the boundary layer thickness, δ is the nominal thickness of the boundary layer, U ∞ is the free stream velocity and τ is the total shear stress.

[0018] In a preferred embodiment, in S2, the wall friction resistance equation including the total shear stress obtained by integrating from α / δ to β / δ along the wall normal is expressed as:

[0019]

[0020] Where β is the distance from the upper limit of the integral to the wall, α is the distance from the lower limit of the integral to the wall, and ρ is the fluid density.

[0021] In a preferred embodiment, in S3, the total shear stress model is expressed as:

[0022]

[0023] Among them, τ + =τ / τ wis the dimensionless total shear stress, is the shear stress at the wall, μ is the fluid dynamic viscosity, H = δ * / θ is the shape factor, δ * is the displacement thickness of the boundary layer, and θ is the momentum thickness of the boundary layer.

[0024] In a preferred embodiment, in S3, the total shear stress model is further subjected to boundary constraints, which are expressed as:

[0025] τ + (y=0)=1.

[0026] In a preferred embodiment, in S4, the wall friction resistance prediction model obtained is expressed as:

[0027]

[0028] Among them, the coefficient B3 can be expressed as:

[0029]

[0030] Wherein, a=0.4945H-2.

[0031] The beneficial effects of the present invention include:

[0032] (1) Significantly reduced dependence on near-wall data;

[0033] (2) Applicable to turbulent boundary layers on smooth and rough surfaces;

[0034] (3) It has the characteristics of high precision, small computational complexity and strong versatility. It can use the flow field information of the outer layer of the boundary layer (average velocity and Reynolds stress) to accurately predict the wall friction resistance, with a prediction error within 5%. BRIEF DESCRIPTION OF THE DRAWINGS

[0035] Figure 1 This is a flow chart of a method for predicting wall friction resistance in the form of momentum integration according to a preferred embodiment of the present invention;

[0036] Figure 2 1 is a comparison chart of the wall friction resistance results obtained in Example 1 and Comparative Example 1;

[0037] Figure 3 is the ratio of the result obtained in Example 2 to the direct numerical simulation result of the smooth wall friction resistance recorded in the corresponding literature;

[0038] Figure 4 is the ratio of the result obtained in Example 3 to the experimental result of smooth wall friction resistance recorded in the corresponding literature;

[0039] Figure 5It is the ratio of the result obtained in Example 4 to the experimental result of rough wall friction resistance recorded in the corresponding literature. DETAILED DESCRIPTION

[0040] 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 apparent.

[0041] 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.

[0042] A method for predicting wall friction resistance in the form of momentum integration provided by the present invention is characterized by comprising the following steps:

[0043] S1. Based on the mean momentum equation of the fluid, the integration method is used to establish the quantitative relationship between the wall friction resistance of the turbulent boundary layer and the turbulence statistics;

[0044] S2. By changing the upper and lower limits of integration and ignoring the difficult-to-obtain near-wall data, the wall friction resistance equation containing the total shear stress is obtained;

[0045] S3. Based on the flow characteristics of the turbulent boundary layer near the wall, a total shear stress model near the wall is proposed;

[0046] S4. Use the near-wall shear stress model to replace the total shear stress in the wall friction drag equation to obtain a wall friction drag prediction model in a zero-pressure-gradient turbulent boundary layer.

[0047] S5. Collect the average velocity and Reynolds stress data of the fluid at the previous position, and obtain the wall friction resistance according to the wall friction resistance prediction model.

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

[0049]

[0050] Among them, the convection term x is the streamwise coordinate, y is the wall normal coordinate, is the average velocity of the fluid flow, is the average normal velocity of the fluid, and is the Reynolds stress, and ν is the fluid kinematic viscosity.

[0051] In S1, the average momentum equation is integrated three times along the normal direction to obtain the quantitative relationship between the wall friction resistance and the turbulence statistics, which is expressed as:

[0052]

[0053] Among them, C f is the wall friction resistance, is the Reynolds number of the boundary layer thickness, δ is the nominal thickness of the boundary layer, U ∞ is the free stream velocity and τ is the total shear stress.

[0054] Furthermore, in S2, the wall friction resistance equation containing the total shear stress obtained by integrating from α / δ to β / δ along the wall normal is expressed as:

[0055]

[0056] Where β is the distance from the upper limit of the integral to the wall, α is the distance from the lower limit of the integral to the wall, and ρ is the fluid density.

[0057] By changing the upper and lower limits of integration, the proposed wall friction equation significantly reduces the dependence of wall friction prediction on near-wall data, thereby resolving the issue of inaccurate wall friction prediction due to the difficulty in acquiring near-wall data. By replacing the streamwise gradient of the average velocity with the wall-normal gradient of the total shear stress, wall friction can be derived by measuring experimental data from only one location in the flow direction.

[0058] 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,

[0059] Furthermore, in S3, the total shear stress model is expressed as:

[0060]

[0061] Among them, τ + =τ / τ w is the dimensionless total shear stress, is the shear stress at the wall, μ is the fluid dynamic viscosity, H = δ * / θ is the shape factor, δ * is the displacement thickness of the boundary layer, and θ is the momentum thickness of the boundary layer.

[0062] Furthermore, the slope of the total shear stress model is related to the shape factor, taking into account the effect of the Reynolds number on the total shear stress. Preferably, in S3, the total shear stress model is also subjected to boundary constraints, which are expressed as:

[0063] τ+ (y=0)=1.

[0064] In S4, the wall friction resistance prediction model obtained is expressed as:

[0065]

[0066] Among them, the coefficient B3 can be expressed as:

[0067]

[0068] Wherein, a=0.4945H-2.

[0069] The wall friction resistance prediction model obtained according to the present invention can accurately predict the wall friction resistance by utilizing the flow field information of the outer layer of the boundary layer. The error of the final wall friction resistance prediction value is within 5%, which can significantly reduce the dependence of the wall friction resistance prediction on the near-wall data.

[0070] Example

[0071] Example 1

[0072] An experiment was conducted to predict the wall friction drag of a turbulent boundary layer, including the following steps:

[0073] S1. Based on the mean momentum equation of the fluid, the integration method is used to establish the quantitative relationship between the wall friction resistance of the turbulent boundary layer and the turbulence statistics;

[0074] S2. By changing the upper and lower limits of integration and ignoring the difficult-to-obtain near-wall data, the wall friction resistance equation containing the total shear stress is obtained;

[0075] S3. Based on the flow characteristics of the turbulent boundary layer near the wall, a total shear stress model near the wall is proposed;

[0076] S4. Use the near-wall shear stress model to replace the total shear stress in the wall friction drag equation to obtain a wall friction drag prediction model in a zero-pressure-gradient turbulent boundary layer.

[0077] S5. Collect the average velocity and Reynolds stress data of the fluid at the previous position, and obtain the wall friction resistance according to the wall friction resistance prediction model.

[0078] In S1, the average momentum equation is:

[0079]

[0080] Among them, the convection term x is the streamwise coordinate, y is the wall normal coordinate, is the average velocity of the fluid flow, is the average normal velocity of the fluid, and is the Reynolds stress, and ν is the fluid kinematic viscosity.

[0081] In S1, the average momentum equation is integrated three times along the normal direction to obtain the quantitative relationship between the wall friction resistance and the turbulence statistics, which is expressed as:

[0082]

[0083] In S2, the wall friction resistance equation including the total shear stress obtained by integrating from α / δ to β / δ along the wall normal is expressed as:

[0084]

[0085] In S3, the total shear stress model is expressed as:

[0086]

[0087] In S3, the total shear stress model is also subjected to boundary constraints, which are expressed as:

[0088] τ + (y=0)=1.

[0089] In S4, the wall friction resistance prediction model obtained is expressed as:

[0090]

[0091] Among them, the coefficient B3 can be expressed as:

[0092]

[0093] Wherein, a=0.4945H-2.

[0094] Among them, the lower limit of the integral is α + =100, and the upper limit of the integral is β / δ=0.4.

[0095] In S5, the wall friction resistance is obtained based on the wall friction resistance prediction model using the data published in existing literature. The literature involved is:

[0096] Literature 1, Schlatter, P., R.Assessment of direct numerical simulationdata of turbulent boundary layers[J].Journal of Fluid Mechanics,2010,659:116-126;

[0097] Document 2. Fernholz, H., Krause, E., Nockemann, M., et al. Comparative measurements in the canonical boundary layer at reδ2≤6×104 on the wall of the german-dutch windtunnel[J]. Physics of Fluids, 1995, 7(6): 1275-1281;

[0098] Document 3. Morrill-Winter, C., Klewicki, J., Baidya, R., et al. Temporally optimized spanwise vorticity sensor measurements in turbulent boundary layers [J]. Experiments in Fluids, 2015, 56(12): 1-14;

[0099] Document 4. Sillero, JA, Jiménez, J., Moser, RD One-point statistics for turbulent wall-bounded flows at reynolds numbers up to δ+≈2000[J]. Physics of Fluids, 2013, 25(10):105102.

[0100] Example 2

[0101] The same experiment as in Example 1 was performed, except that in S5, the wall friction resistance was obtained according to the wall friction resistance prediction model using data from the following literature on direct numerical simulation of turbulent boundary layers on smooth walls. The literature involved is:

[0102] Document 5. Wu,

[0103] Literature 6, Schlatter, P., R.Assessment of direct numerical simulationdata of turbulent boundary layers[J].Journal of Fluid Mechanics,2010,659:116-126;

[0104] Document 7. Jiménez, J., Hoyas, S., Simens, MP, et al. Turbulent boundary layers and channels at moderate reynolds numbers [J]. Journal of Fluid Mechanics, 2010, 657: 335-360;

[0105] Document 8. Sillero, JA, Jiménez, J., Moser, RD One-point statistics for turbulent wall-bounded flows at reynolds numbers up to δ+≈2000[J]. Physics of Fluids, 2013, 25(10):105102.

[0106] Example 3

[0107] The same experiment as in Example 1 was performed, except that, in S5, the wall friction resistance was obtained according to the wall friction resistance prediction model using data from the following literature on smooth wall turbulent boundary layer experiments:

[0108] Literature 9. Efros, V., Krogstad, P.- Development of turbulent boundarylayer after a step from smooth to rough surface[J].Experiments in fluids,2011,51(6):1563-1575;

[0109] Document 10 J.M. Experimental studies of zero pressure-gradient turbulent boundary layer flow[D]. Stockholm: KTH Royal Institute of Technology, 1999;

[0110] Reference 11. Harun, Z. The structure of adverse and favourable pressure gradient turbulent boundary layers[D]. Melbourne: University of Melbourne, 2012;

[0111] Reference 12. Harun, Z., Monty, J.P., Mathis, R., et al. Pressure gradient effects on the large-scale structure of turbulent boundary layers[J]. Journal of Fluid Mechanics, 2013, 715: 477 - 498;

[0112] Reference 13. Erm, L.P., Joubert, P.N. Low-reynolds-number turbulent boundary layers[J]. Journal of Fluid Mechanics, 1991, 230: 1 - 44;

[0113] Reference 14. Fernholz, H., Krause, E., Nockemann, M., et al. Comparative measurements in the canonical boundary layer at reδ2 ≤ 6×104 on the wall of the german-dutch windtunnel[J]. Physics of Fluids, 1995, 7(6): 1275 - 1281;

[0114] Reference 15. Balint, J.-L., Wallace, J.M., P. The velocity and vorticity vector fields of a turbulent boundary layer. Part 2. Statistical properties[J]. Journal of fluid mechanics, 1991, 228: 53 - 86;

[0115] Document XVI. Smith, R. W. Effect of reynolds number on the structure of turbulent boundary layers[D]. Princeton: Princeton University, 1994;

[0116] Document XVII. Schultz, M. P., Flack, K. A. Turbulent boundary layers over surfaces smoothed by sanding[J]. J. Fluids Eng., 2003, 125(5): 863 - 870;

[0117] Document XVIII. Schultz, M., Flack, K. Outer layer similarity in fully rough turbulent boundary layers[J]. Experiments in fluids, 2005, 38(3): 328 - 340;

[0118] Document XIX. Talluru, K., Baidya, R., Hutchins, N., et al. Amplitude modulation of all three velocity components in turbulent boundary layers[J]. Journal of Fluid Mechanics, 2014, 746;

[0119] Document XX. Morrill - Winter, C., Klewicki, J., Baidya, R., et al. Temporally optimized spanwise vorticity sensor measurements in turbulent boundary layers[J]. Experiments in Fluids, 2015, 56(12): 1 - 14.

[0120] Example 4

[0121] The same experiment as in Example 1 was performed, except that in S5, the wall friction resistance was obtained according to the wall friction resistance prediction model using data from the following literature on rough wall turbulent boundary layer experiments:

[0122] Literature 21, Efros, V., Krogstad, P.- Development of turbulent boundarylayer after a step from smooth to rough surface[J].Experiments in fluids,2011,51(6):1563-1575;

[0123] Document 22. Morrill-Winter, C., Squire, D., Klewicki, J., etal. Reynoldsnumber and roughness effects on turbulent stresses in sandpaper roughnessboundary layers[J]. Physical Review Fluids, 2017, 2(5):054608;

[0124] Reference 23. Saddoughi, SG, Veeravalli, SV Local isotropy in turbulent boundary layers at high reynolds number [J]. Journal of Fluid Mechanics, 1994, 268: 333-372;

[0125] Document 24. Brzek, B., Cal, RB, Johansson, G., et al. Inner and outerscalings in rough surface zero pressure gradient turbulent boundary layers [J]. Physics of Fluids, 2007, 19(6):065101;

[0126] Document 25. Mehdi, F., Klewicki, J., White, C. Mean force structure and its scaling in rough-wall turbulent boundary layers [J]. Journal of Fluid Mechanics, 2013, 731: 682-712.

[0127] Comparative Example 1

[0128] The same experiment as in Example 1 was performed, except that the wall friction resistance was obtained using the total shear stress model method proposed by Kumar.

[0129] The detailed process of the total shear stress model proposed by Kumar can be found in the literature Kumar, P., Mahesh, K. A method to determine wall shear stress from mean profiles in turbulent boundary layers[J]. Experiments in Fluids, 2022, 63(1):6.

[0130] Taking the wall friction resistance recorded in Documents 1 to 4 as the actual value, the wall friction resistance obtained in Example 1 and Comparative Example 1 is compared. The results are as follows: Figure 2 As shown, the horizontal axis is the momentum thickness Reynolds number, U ∞ is the free flow velocity, θ is the momentum thickness of the flat boundary layer, and ν is the fluid kinematic viscosity; the ordinate is the wall friction resistance error; it can be seen from the figure that the wall friction resistance error obtained by the method in Example 1 is smaller than the wall friction resistance error obtained by the total shear stress model method proposed by Kumar in Comparative Example 1.

[0131] References 5 to 8 record the direct numerical simulation data of smooth wall friction resistance, and the results obtained in Example 2 are recorded as C f,p , the smooth wall friction resistance recorded in the literature is recorded as C f,t , C f,p with C f,t The ratio of Figure 3 As shown in the figure, it can be seen that the difference between the results obtained in Example 2 and the results obtained in other literature is less than 5%, indicating that the prediction error in Example 2 is within 5%.

[0132] The experimental data of friction resistance of smooth wall are recorded in Documents 9 to 20. The results obtained in Example 3 are recorded as C f,p, the smooth wall friction resistance recorded in the literature is recorded as C f,t , C f,p with C f,t The ratio of Figure 4 As shown in the figure, it can be seen that the difference between the results obtained in Example 3 and the results obtained in other literature is less than 5%, indicating that the prediction error in Example 3 is within 5%.

[0133] References 21 to 25 record the experimental data of rough wall friction resistance. The results obtained in Example 4 are recorded as C f,p , the rough wall friction resistance recorded in the literature is recorded as C f,t , C f,p with C f,t The ratio of Figure 5 As shown in the figure, it can be seen that the difference between the results obtained in Example 4 and the results obtained in other literature is less than 5%, indicating that the prediction error in Example 4 is within 5%.

[0134] In the description of the present invention, it should be noted that the terms "upper," "lower," "inner," "outer," "front," and "rear" and the like, indicating positions or locations, are based on the operating state of the present invention and are intended solely to facilitate and simplify the description of the present invention. They are not intended to indicate or imply that the devices or components referred to must have, be constructed, or operate in a specific orientation, and therefore should not be construed as limitations on the present invention. Furthermore, the terms "first," "second," "third," and "fourth" are used for descriptive purposes only and should not be construed as indicating or implying relative importance.

[0135] In the description of the present invention, it should be noted that, unless otherwise expressly specified or limited, the terms "mounted," "connected," and "connected" should be understood in a broad sense. For example, they can refer to fixed, detachable, or integral connections; mechanical or electrical connections; direct or indirect connections through an intermediate medium; and internal communication between two components. Those skilled in the art will understand the specific meanings of the above terms in the present invention in specific contexts.

[0136] 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 method for predicting wall friction resistance in the form of momentum integration, characterized in that: The following steps are involved: S1. Based on the mean momentum equation of the fluid, the integration method is used to establish the quantitative relationship between the wall friction resistance of the turbulent boundary layer and the turbulence statistics; S2. By changing the upper and lower limits of integration and ignoring the difficult-to-obtain near-wall data, the wall friction resistance equation containing the total shear stress is obtained; S3. Based on the flow characteristics of the turbulent boundary layer near the wall, a total shear stress model near the wall is proposed; S4. Use the near-wall shear stress model to replace the total shear stress in the wall friction drag equation to obtain a wall friction drag prediction model in a zero-pressure-gradient turbulent boundary layer. S5. Collect the average velocity and Reynolds stress data of the fluid at the previous position, and obtain the wall friction resistance according to the wall friction resistance prediction model. In S1, the average momentum equation is: Among them, the convection term x is the streamwise coordinate, y is the wall normal coordinate, is the average velocity of the fluid flow, is the average normal velocity of the fluid, and is the Reynolds stress, ν is the fluid kinematic viscosity, In S1, the average momentum equation is integrated three times along the normal direction to obtain the quantitative relationship between the wall friction resistance and the turbulence statistics, which is expressed as: Among them, C f is the wall friction resistance, is the Reynolds number of the boundary layer thickness, δ is the nominal thickness of the boundary layer, U ∞ is the free flow velocity, τ is the total shear stress, In S2, the wall friction resistance equation including the total shear stress obtained by integrating from α / δ to β / δ along the wall normal is expressed as: Among them, β is the distance from the upper limit of integration to the wall, α is the distance from the lower limit of integration to the wall, ρ is the fluid density, In S3, the total shear stress model is expressed as: Among them, τ + =τ / τ w is the dimensionless total shear stress, is the shear stress at the wall, μ is the fluid dynamic viscosity, H = δ * / θ is the shape factor, δ * is the boundary layer displacement thickness, θ is the boundary layer momentum thickness, In S4, the wall friction resistance prediction model obtained is expressed as: Among them, the coefficient B3 can be expressed as: Wherein, a=0.4945H-2.

2. The wall friction resistance prediction method in momentum integral form according to claim 1 is characterized in that: In S3, the total shear stress model is also subjected to boundary constraints, which are expressed as: t + (y=0)=1.

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