A method for predicting the thermal-acoustic coupling effect of high-speed flow in wall-cooled scenarios

Abstract

This invention discloses a method for predicting the thermo-fluid-acoustic coupling effect in high-speed flow scenarios with wall cooling, relating to the fields of fluid mechanics and aerospace technology. The method includes: S1, obtaining the secondary peak position from the pulsating pressure normal distribution curve based on existing simulation databases, which characterizes the sensitivity of incoming flow conditions and wall thermal state; S2, selecting multiple characteristic dimensionless numbers related to wall temperature and conducting correlation analysis between these characteristic dimensionless numbers and the wall temperature to obtain the typical dimensionless number with the strongest correlation to the secondary peak position of the pulsating pressure distribution; S3, characterizing the variation law of the wall temperature; and S4, using this variation law to predict the wall temperature in the prediction of the thermo-fluid-acoustic coupling effect under high-speed flow conditions. This invention clarifies the quantitative correlation of multi-field coupling between flow, thermo, and acoustics, providing theoretical support for engineering design and experiments.

Description

TECHNICAL FIELD

[0001] The present application relates to the field of fluid mechanics and aerospace technology. More particularly, the present application relates to a method for predicting high-speed flow heat flux acoustic coupling effect in wall cooling scenario. BACKGROUND

[0002] In aircraft design, the fluctuating pressure of compressible turbulent boundary layer directly affects the structural vibration and aerodynamic noise, and is a key factor determining the performance and safety of aircraft. Due to the strong aerodynamic heating of the flow field at high Mach number and the strong wall heat transfer, it cannot be directly simplified as an adiabatic wall condition, and the wall cooling effect should be considered. Existing research shows that when the wall is cooled, the temperature, density distribution and turbulent structure of the flow field will change significantly, which in turn leads to changes in the normal distribution characteristics of the fluctuating pressure (such as peak position, amplitude). However, the core dimensionless number of the fluctuating pressure of the compressible turbulent boundary layer affected by the wall cooling has not been fully clarified, and there is also a lack of systematic explanation of the dominant mechanism, making it difficult to quantitatively predict the fluctuating pressure characteristics under cooling conditions in engineering design.

[0003] Existing research has made a detailed discussion on the influence of wall cooling on the mass and energy transport of compressible turbulent boundary layer, but there is still no work directly relating the change of wall temperature to the fluctuating pressure. The reduction of wall temperature intensifies the isotropization of near-wall temperature fluctuation, and the temperature fluctuation structure of near-wall slender stripes gradually collapses, but the fixed Flow structures are similar at different Mach numbers M (Cogo M, Baù U, Chinappi M, et al. Assessment of heattransfer and Mch number effects on highspeed turbulent boundary layers[J]. Journal of Fluid Mechanics, 2023, 974:A10). Compared to M, the buffer layer and the logarithmic layer affect the turbulent kinetic energy (TKE) (Tao et al., 2020). The wall cooling dominates the energy exchange in the near-wall region (Huang, J., Duan, L. & Choudhari, M. M. 2022 Direct numerical simulation of hypersonic turbulent boundary layers: effect of spatial evolution and Reynolds number. J. Fluid Mech. 937, A3). The wall cooling can also suppress the separation of the turbulent size scale caused by the increase of M (Fan, Y., Li, W. & Pirozzoli, S. 2022 Energy exchanges in hypersonic flows. Phys. Rev. Fluids 7(9), L092601). Gibis et al. (Gibis T, Sciacovelli L, Kloker M, et al. Heat-transfer effects in compressible turbulent boundary layers - a regime diagram[J]. Journal of Fluid Mechanics, 2024) classified the flow into heating, adiabatic, or weak / medium / strong / quasi-incompressible cooling conditions according to the wall-normal position of the temperature peak of the boundary layer at different Eckert numbers . This classification can be obtained from the prior estimation of E C and the local Reynolds number ( is the friction velocity, is the boundary layer thickness, is the wall skin friction coefficient) effectively estimates the degree of heat transfer effect in a given compressible turbulent boundary layer. With the increase of wall cooling intensity, the temperature peak gradually moves away from the wall, and the interaction between the region with positive temperature gradient and the turbulent boundary layer significantly increases, which indicates that the temperature peak is a decisive factor of the non-adiabatic boundary layer topological characteristics.

[0004] Regarding the influence of wall cooling on the fluctuating pressure, Zhang et al. (Full name: Zhang Y, Li X, Liu X, et al. Wall-cooling effects on pressure fluctuations in compressible turbulent boundary layers from subsonic to hypersonic regimes[J]. Journal of Fluid Mechanics, 2022, 946: A31) extended the fluctuating pressure decomposition method of Pope (Full name: Pope S B. Turbulent Flows [M]. Cambridge University Press, 2000) for incompressible flow to compressible turbulent boundary layers, and divided the fluctuating pressure in the compressible turbulent boundary layer into fast pressure, slow pressure, compressible pressure, viscous pressure and harmonic pressure. Combined with the results of direct numerical simulation, it is found that wall cooling can enhance the compressible pressure and suppress other components, and at high Mach number M=8, the proportion of this enhancement even exceeds 100%. This shows that the pressure fluctuation is generated by vortex mode and acoustic mode, and both modes are closely related to compressibility. In the case of subsonic, supersonic and hypersonic, wall cooling suppresses the vortex mode of pressure fluctuation in the boundary layer, but enhances the acoustic mode near the wall. However, this pressure decomposition method is based on the continuity equation and momentum equation, and does not explicitly introduce the influence of temperature.

[0005] Based on the above discussion, the current research on the characteristic parameters of the influence of wall temperature on fluctuating pressure is not complete, and the mechanism is not clear enough. Therefore, it is urgent to establish an analysis method based on characteristic dimensionless numbers to reveal the influence law of wall cooling on the normal distribution of fluctuating pressure, which can be used in the prediction of thermal-acoustic coupling effect in high-speed flow environment or the analysis and structure optimization of fluctuating pressure characteristics. SUMMARY

[0006] An object of the present application is to solve at least the above problems and / or defects, and to provide at least the advantages to be described later.

[0007] In order to achieve these objects and other advantages of the present application, a method for predicting thermal-acoustic coupling effect in high-speed flow in the face of wall cooling scene is provided, comprising:

[0008] S1, obtaining the secondary peak position from the fluctuating pressure normal distribution curve based on the existing simulation database, which can represent the sensitivity of the incoming flow condition and the wall surface heat state ;

[0009] S2, selecting a plurality of characteristic dimensionless numbers related to the wall surface temperature for the problem of the influence of the wall surface temperature and the Mach number on the fluctuating pressure secondary peak position, and carrying out correlation analysis of the dimensionless numbers and to obtain the typical dimensionless number with the strongest correlation ;

[0010] S3, representing the change rule of with the change of :

[0011] ;

[0012] In the above formula, is the Eckert number;

[0013] S4, in the prediction of the heat-flow-sound coupling effect in the high-speed flow environment, the change rule of S3 is introduced to complete the prediction of .

[0014] Preferably, in S1, the database is high-fidelity flow field data obtained by direct numerical simulation DNS.

[0015] Preferably, the abscissa of the fluctuating pressure normal distribution curve uses the normal coordinate of the inner scale , and the ordinate is the value of non-dimensionalized using the incoming flow average pressure ;

[0016] The secondary peak position systematically deviates with the change of the Mach number and the wall surface temperature.

[0017] Preferably, in S2, the plurality of characteristic dimensionless numbers are obtained in the following manner:

[0018] The total enthalpy form of the compressible energy equation is non-dimensionalized, and when the non-dimensional pressure term coefficient is 1, the characteristic dimensionless number related to the wall surface temperature is selected from the equation: , , M, in combination with and , wherein M is the incoming flow Mach number, is the ratio of the wall surface temperature to the recovery temperature, is the non-adiabatic parameter.

[0019] Preferably, in S2, the dimensionless number correlation analysis includes:​

[0020] S20, calculating characteristic dimensionless numbers under different working conditions, ;

[0021] S21, obtaining the Pearson correlation coefficient between the plurality of characteristic dimensionless numbers and the peak position of the fluctuating pressure;

[0022] S22, using the Pearson correlation coefficient as a quantitative screening basis to screen out the strongest correlation in the wall cooling influence as a typical dimensionless number.

[0023] The present application at least includes the following beneficial effects: the present application aims at the problem that wall cooling leads to significant changes in the fluctuating pressure distribution of the compressible turbulent boundary layer flow field, and proposes a method for predicting the thermo-acoustic coupling effect in high-speed flow environment. The method determines the key dimensionless number-Eckert number (Ec) through dimension analysis and correlation analysis method, and gives the influence law of Ec on the normal peak position of fluctuating pressure based on direct numerical simulation data; the influence mechanism is analyzed by referring to the pressure decomposition method of Zhang et al. (2022). The present application clearly quantifies the correlation of flow-thermal-acoustic multi-field coupling, and the influence law of the dimensionless number proposed by the present application has the potential to more complex geometric configurations and flow field analysis, providing theoretical support for engineering design and experiment.

[0024] Other advantages, objects and features of the present application will be partly embodied by the following description, and partly understood by those skilled in the art through research and practice of the present application. BRIEF DESCRIPTION OF DRAWINGS

[0025] Figure 1 Fig. 1 is the fluctuating pressure normal distribution curve of four working conditions in Table 1;

[0026] Figure 2 Fig. 3 is a graph showing the variation of the position of the secondary peak with under different working conditions.

[0027] Figure 3 Fig. 4 is a residual distribution graph of the prediction model for the reference data and the verification data. DETAILED DESCRIPTION

[0028] The present application will be further described in detail below with reference to the accompanying drawings, so that those skilled in the art can implement it according to the description and the drawings.

[0029] ​​A method for predicting high-speed flow thermal flow acoustic coupling effect in wall cooling scene, proposes to take the secondary peak position of fluctuating pressure distribution as the key feature of the normal distribution of fluctuating pressure affected by wall cooling, through systematic dimension analysis and data analysis, identifies Eckert number (Ec) as the key dimensionless parameter for the first time to characterize the normal distribution of fluctuating pressure affected by wall cooling. Specifically, by introducing the Pearson correlation coefficient to quantitatively screen the candidate parameters (such as M, Tw / Tr, 1 / Ec, etc.), the correlation coefficients of each parameter with the secondary peak position are calculated (0.2877, -0.8846, 0.8897, -0.8880, 0.5874, respectively), and based on the principle of maximum absolute value of correlation coefficient (0.8897), 1 / Ec is selected as the core parameter. This method gives the core parameter of the secondary peak position of fluctuating pressure affected by wall cooling on the basis of various dimensionless numbers proposed in existing research, provides quantifiable screening basis for engineering prediction, significantly improves analysis efficiency, and lays a parameterized foundation for subsequent modeling, including the following steps:

[0030] S1, obtain the typical characteristics (i.e. secondary peak position ) of the fluctuating pressure distribution of compressible turbulent boundary layer;

[0031] In this step, the high-fidelity direct numerical simulation (Direct Numerical Simulation, DNS) data disclosed by Zhang (2018) et al. is used, which will also be used as the verification data of the model in S4 (paper title: Direct Numerical Simulation Database for Supersonic and Hypersonic Turbulent Boundary Layers. Database website: https: / / turbmodels.larc.nasa.gov / Other_DNS_Data / supersonic_hypersonic_flatplate.html), the data obtained by DNS method has high precision, which can accurately obtain the normal fluctuating pressure distribution of supersonic / hypersonic turbulent boundary layer.

[0032] The free stream Mach number and wall temperature conditions of DNS examples under different working conditions are shown in Table 1, the first column of Table 1 is the working condition name, the number after M represents the Mach number, and the number after T represents the ratio of wall temperature to recovery temperature.

[0033] Table 1

[0034]

[0035] Taking the data in Table 1 as an example, Figure 1Table 1 shows the normal distribution curves of boundary layer pulsating pressure for the four operating conditions. The horizontal axis uses the normal coordinates with an internal scale. ,and (Right now It is the physical distance y normal to the wall and the viscous length scale. The ratio of the two values ​​(the vertical axis represents the average pressure of the incoming flow) is shown on the horizontal axis. Dimensionless Value (i.e.) ),from Figure 1 It can be seen that under all operating conditions Follow The increase in all values ​​shows a non-monotonic trend, and the Mach number and wall temperature ratio have a significant impact on the overall intensity of pulsating pressure: the working condition with a high wall temperature ratio and a low Mach number exhibits a lower intensity of pulsating pressure.

[0036] Furthermore, under the three conditions with high cooling intensity, the pulsating pressure exhibits a first peak at the wall surface, which, according to the conclusions of Zhang et al. (2022), is caused by near-wall expansion at high Mach numbers; in all four conditions, there is a second pressure peak roughly located in the logarithmic region of the boundary layer. Figure 1 The peaks are marked with dots and are referred to as "secondary peaks" below.

[0037] Additionally, in the area adjacent to the wall (very small) While the first peak may appear under conditions with high cooling intensity (such as M6T025 and M14T018), this peak does not consistently occur under all conditions and its existence is highly conditionally dependent. Therefore, it is not suitable as a universal characteristic of pulsating pressure distribution. In contrast, the second peak, located in the logarithmic region, is clearly visible under all conditions, and its normal position... (Right now Figure 1 The abscissas corresponding to the positions of the secondary peaks of each curve in the figure show a systematic shift as the Mach number and wall temperature change, indicating that the position of the secondary peak is very sensitive to both the incoming flow conditions and the thermal state of the wall.

[0038] Therefore, the second peak is the key, stable, and identifiable feature that truly reflects the combined influence of wall temperature and incoming Mach number. Based on this assessment, the subsequent work of this invention will focus on the quantitative analysis and modeling of this second peak location to reveal the mechanism by which wall cooling affects the pulsating pressure distribution.

[0039] S2. By dimensionlessly transforming the energy equation and combining it with previous research, several characteristic dimensionless numbers related to wall temperature were extracted, and the relationship between these characteristic dimensionless numbers and... Correlation analysis was performed to obtain the typical dimensionless coefficients with the strongest correlation between pulsating pressure distribution and wall cooling effect. .

[0040] This step uses the typical dimensionless number correlation analysis method to obtain the typical dimensionless number that has the strongest correlation with the wall cooling and pulsating pressure distribution. The specific method is as follows:

[0041] To clarify the typical dimensionless parameters that affect the position of the secondary peak due to wall cooling, the compressible energy equation is first made dimensionless.

[0042] The compressible energy equation in total enthalpy form is:

[0043] ;

[0044] In the above formula, Let be the density, and t be the time. For total enthalpy, For heat transfer, The rate of change of pressure over time. It is a viscous force. This is the velocity vector.

[0045] Dimensionless transformation of the compressible energy equation yields:

[0046] ;

[0047] In the above formula, the superscript "*" indicates a dimensionless quantity. For specific heat ratio, v is the flow velocity, v is the normal velocity, and y is the normal coordinate. The outer edge temperature of the boundary layer. The wall temperature, Thermal conductivity, For the incoming flow velocity, Let M be the incoming flow density and M be the incoming flow Mach number.

[0048] For Ekert numbers, The outer edge temperature of the boundary layer (approximately equal to the incoming flow temperature) ), This refers to the adiabatic wall temperature (i.e., the recovery temperature). Let be the wall temperature, and r be the recovery factor. Where c is the boundary layer thickness and c is the speed of sound. The time is the characteristic of the flow.

[0049] When the coefficient of the dimensionless pressure term is 1, the term related to the wall temperature can be selected from the equation. , These are the characteristic parameters, M, etc. In addition, the characteristic parameters commonly used in the literature include... , Since Cogo et al. (2023) have already pointed out... Equivalent to Ec ( ), here we will no longer mention To be taken into consideration.

[0050] To ensure the universality and reliability of the correlation analysis results, the selected data sample must fully cover the variation range of key parameters in actual engineering, and its distribution should have a certain degree of uniformity to avoid reducing the validity of the conclusions due to data concentration or bias. This part of the data was provided by Professor Zhang Pengjunyi of the University of Science and Technology of China. To ensure the reliability of comparisons between different working conditions, following the common practice in current research, the friction Reynolds number at the flow direction position of each working condition was set as follows. All are 550. Calculate the four characteristic dimensionless numbers and the position of the normal secondary peak of the pulsating pressure under each operating condition. As shown in Table 2:

[0051] Table 2

[0052]

[0053] The last row of Table 2 gives the dimensions of each dimensionless number and The Pearson correlation coefficient between them. This is used to select the parameters most relevant to the target performance from multiple candidate parameters. The most correlated typical parameter set was selected, and then the typical dimensionless numbers that affect the secondary peak position of the normal pulsating pressure distribution due to wall cooling were screened out. This invention uses the Pearson correlation coefficient as a quantitative screening basis, which is used to accurately measure the strength and direction of the linear correlation between two sets of data. This is achieved by identifying the secondary peak positions of the pulsating pressure under various operating conditions. The four sets of candidate independent variable data were input into Excel software, and the CORREL function was used to calculate their corresponding Pearson correlation coefficients, which were -0.14244, -0.86704, 0.888035, and 0.584076, respectively. The analysis results show that the third set of parameters has the largest absolute value of the correlation coefficient (0.888035), indicating that its linear correlation is the most significant. Based on this statistical analysis conclusion, this invention ultimately selected the third set of parameters, 1 / Ec, which has the strongest correlation, as the typical dimensionless number.

[0054] S3, Quantitative Description The variation pattern with 1 / Ec;

[0055] This step, based on a high-fidelity database with a large-scale uniformly distributed parameter space, yields a formula through curve fitting to characterize... The variation pattern with 1 / Ec is as follows:

[0056] To clarify the quantitative relationship between the secondary peak position of the pulsating pressure distribution and the Eckert number (Ec), this invention proposes a method based on the dimensionless parameter 1 / Ec to determine the secondary peak position of the normal distribution of pulsating pressure. Prediction methods.Figure 2 The table in Table 2 is shown. The variation of 1 / Ec is shown, where the horizontal axis represents 1 / Ec and the vertical axis represents the position of the secondary peak. .like Figure 2 As shown, It shows a significant positive correlation with 1 / Ec. Through precise curve fitting of the data using MATLAB, the two points with the largest 1 / Ec values ​​show significant dispersion, possibly caused by other errors; these points were discarded during the fitting process, thus obtaining the desired result. Figure 2 The fitted curve in red has the following fitting equation: ;

[0057] The maximum error between the above fitting formula and the fitted data is within 15%, which can describe the data well. The relationship between 1 / Ec and the established quantitative relationship provides an important basis for modeling and analysis of pulsating pressure characteristics and structural optimization in engineering applications under high-speed flow environments. In other words, the above relationship can provide a basis for predicting the pulsating pressure distribution under different incoming flow and wall temperature conditions, and provide a reference for modeling research on the influence of wall temperature on pulsating pressure distribution.

[0058] S4. In the prediction of the thermal-fluid-acoustic coupling effect under high-speed flow conditions, the variation law of S3 is introduced to complete the prediction of 1 / Ec. The prediction.

[0059] Data on the normal distribution of pulsating pressure under different operating conditions have been provided in S1. These data will be used to verify the patterns obtained in S3. Extraction Figure 1 The location of the secondary peak of the mid-pulsation pressure and the corresponding Ec, such as Figure 2 As shown in the four points referred to in the "new data" section, the relative errors in the predictions for the new data are all within 10%, indicating that the fitted model has predicted the positions of the secondary peaks of pulsating pressure well under different operating conditions. Figure 3 Furthermore, the residuals between the fitted model and the actual values ​​are given ( Figure 3 The residual of the middle vertical axis is Figure 2 The difference between the actual and fitted values ​​of the secondary peak positions of the pulsating pressure under various operating conditions demonstrates the good performance of the model.

[0060] Therefore, the law given by S3 has good universality. The following is a mechanism exploration based on the pulsating pressure decomposition method proposed by Zhang et al. (2022):

[0061] Zhang et al. (2022) derived the normal distribution of the total pulsating pressure and each pulsating pressure component under different operating conditions by solving the pressure equations for each component. This is used to demonstrate the relationship between different Mach numbers (M) and wall temperature ratios (M / M). Tw / TrUnder the operating condition, the total pulsating pressure and the normal distribution characteristics of its components are analyzed. By comparing the pressure component curves under adiabatic and cooling conditions at the same Mach number, it can be clearly found that only the slow pressure component ( ps A significant peak value is observed at the location of the secondary peak of the total pulsating pressure, and this peak value shifts significantly outward under the effect of wall cooling. Further analysis shows that the location of the slow pressure peak value in each operating condition highly coincides with the secondary peak of the total pulsating pressure, confirming that the secondary peak phenomenon in the normal distribution of pulsating pressure mainly originates from the contribution of the slow pressure component. Slow pressure physically characterizes the intensity of the nonlinear interaction between turbulent pulsations; therefore, it can be concluded that wall cooling affects the position of the secondary peak of pulsating pressure in the normal distribution by modulating the nonlinear interaction mechanism of turbulent pulsations. This discovery not only provides a deeper mechanistic explanation for the influence of the Eckert number (Ec) on the position of the secondary peak, but also deepens the understanding of the influence of wall cooling on the distribution behavior of pulsating pressure, and has important theoretical and engineering significance for the prediction and control of the thermal-fluid-acoustic coupling effect in high-speed flow environments.

[0062] This invention is the first to combine the Ec classification method of Gibis (2024) (such as weak cooling and medium cooling conditions) with pulsating pressure analysis, and establishes a causal chain of "cooling intensity → turbulent nonlinear effect → secondary peak position". By deepening the understanding of the flow-heat-acoustic multi-field coupling mechanism, it provides theoretical support for the design of thermal protection system and aerodynamic noise control of high-speed aircraft. The main effects are: (1) universality: the dimensionless number analysis method has the potential to be extended to complex geometric configurations (such as curved wall boundary layer), which can provide a reference for engineering design; (2) efficiency improvement: the prediction efficiency is improved by quantitative relationship and the experimental cost is reduced; (3) closed loop of technological innovation: a complete analysis chain is formed from parameter screening to mechanism explanation, providing new ideas for pulsating pressure analysis and noise optimization.

[0063] The above solution is merely an illustration of a preferred example and is not limited thereto. When implementing this invention, appropriate substitutions and / or modifications can be made according to the user's needs.

[0064] Although embodiments of the present invention have been disclosed above, they are not limited to the applications listed in the specification and embodiments. It can be applied to various fields suitable for the present invention. Other modifications can be readily made by those skilled in the art. Therefore, without departing from the general concept defined by the claims and their equivalents, the present invention is not limited to the specific details and examples shown and described herein.

Claims

1. A method for predicting the thermal-acoustic coupling effect of high-speed flow in a wall-cooling scenario, characterized in that, The application relates to a method for predicting the flow field of a hypersonic vehicle, comprising the following steps: S1, based on the existing simulation database, obtaining the secondary peak position from the fluctuating pressure normal distribution curve, which can represent the sensitivity of the incoming flow conditions and the wall heat state ; S2, for the problem of the influence of wall temperature and Mach number on the position of the secondary peak of fluctuating pressure, a plurality of characteristic dimensionless numbers related to wall temperature are selected, and correlation analysis is carried out between the dimensionless numbers and to obtain the typical dimensionless number with the strongest correlation. S3, Characterization by curve fitting With the change law: ； In the above formulae, is the Eötvös number; S4, in the prediction of thermo-aero-acoustic coupling effect under high speed flow environment, the variation law of S3 is introduced to complete to prediction.

2. The method of claim 1, wherein the wall-cooled scenario is characterized by, In S1, the simulation database is high-fidelity flow field data obtained by using a direct numerical simulation method (DNS).

3. The method of claim 1, wherein the wall-cooled scenario is characterized by, The abscissa of the curve of the normal distribution of the pulsatile pressure uses the normal coordinate of the inner scale , the ordinate is the dimensionless value of the use of the average pressure of the incoming flow ; the dimensionless value of the use of the average pressure of the incoming flow ; The normal position at the position of the secondary peak position is systematically deviated with the change of the Mach number and the wall temperature.

4. The method of claim 1, wherein the wall-cooled scenario is characterized by, In S2, the acquisition mode of the plurality of characteristic dimensionless numbers is as follows: The total enthalpy form of compressible energy equation is non-dimensionalized, and the characteristic non-dimensional number related to the wall temperature is selected from the equation when the non-dimensional pressure term coefficient is 1: , , M, combined with and , where M is the Mach number of the incoming flow, is the ratio of the wall temperature to the recovery temperature, is the non-adiabatic parameter.

5. The method of claim 1, wherein the wall-cooled scenario is characterized by, In S2, the correlation analysis mode comprises: S20, calculating the characteristic dimensionless number under different working conditions, ; S21, obtaining a Pearson correlation coefficient between the plurality of characteristic dimensionless numbers and the plurality of characteristic dimensionless numbers. S22, the Pearson correlation coefficient is used as a quantitative screening basis to screen out the most relevant wall cooling effect as a typical dimensionless number.​

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