Particle image velocimetry out-of-plane velocity measurement method

Abstract

The invention relates to the technical field of fluid mechanics measurement, solves the technical problems of obvious out-of-plane velocity error and poor adaptability to complex flow scenes when a linear displacement model is adopted to invert a three-dimensional velocity field in a traditional method, and particularly relates to an out-of-plane velocity measurement method for particle image velocity measurement. Comprising the following steps: constructing a nonlinear displacement model containing a second-order Hessian item based on a displacement relationship between an image plane and an object space; establishing a residual function and an optimization objective function; performing iterative optimization solution on the residual function by taking the optimization objective function as an objective to obtain object space displacement; and calculating a three-dimensional velocity field according to the object space displacement. According to the method, the non-linear displacement model containing the second-order Hessian item is constructed, the coupling effect of out-of-plane displacement and image distortion is accurately described, and in a scene with high out-of-plane speed or facing a strong shearing and vortex flow field, the stability of errors can be ensured, so that the calculation precision of the out-of-plane speed is remarkably improved.

Description

Technical Field

[0001] The present invention relates to the technical field of fluid mechanics measurement, and particularly to an out-of-plane velocity measurement method for particle image velocimetry. Background Art

[0002] Traditional particle image velocimetry (PIV) technology analyzes the apparent displacement of a pair of particle images and uses first-order Taylor expansion to establish a linear displacement model to invert the three-dimensional velocity field. However, the first-order model ignores higher-order non-linear terms, resulting in the following problems:

[0003] 1. Significant out-of-plane velocity error: The out-of-plane displacement (Δz) causes image scaling and distortion, and its non-linear effect cannot be accurately described by the linear model at large displacements.

[0004] 2. Poor adaptability to complex flows: In strong shear flow or vortex flow fields, the influence of higher-order terms of the velocity gradient on the apparent displacement is exacerbated, and the accuracy of the linear model decreases.

[0005] Existing improvement methods mostly rely on multi-camera calibration or increasing the illumination thickness compensation, but the system complexity is high and the cost is expensive. Specifically, the out-of-plane velocity error is attributed to physical hardware and working conditions, and the influence caused by the light field is solved by piling up hardware devices. Increasing the illumination thickness will result in more background noise, and further requires a more complex calibration process. Therefore, when implementing this method, it is necessary to balance the improvement of illumination intensity and the signal-to-noise ratio, and design reasonable experiments to verify its effectiveness. High-precision equipment is required, so it is not easy to use and the cost is expensive. Summary of the Invention

[0006] Aiming at the deficiencies of the prior art, the present invention provides an out-of-plane velocity measurement method for particle image velocimetry, which solves the technical problems of significant out-of-plane velocity error and poor adaptability to complex flow scenarios when the traditional method uses a linear displacement model to invert the three-dimensional velocity field.

[0007] To solve the above technical problems, the present invention provides the following technical solutions: An out-of-plane velocity measurement method for particle image velocimetry, the method comprising the following steps:

[0008] S1. Construct a non-linear displacement model containing the second-order Hessian term based on the displacement relationship ΔX in the image plane and the object space;

[0009] S2. Establish a residual function Loss(Δx) and an optimization objective function E(Δx) for optimizing the non-linear displacement model;

[0010] S3. Iteratively optimize and solve the residual function Loss(Δx) with the optimization objective function E(Δx) as the target to obtain the object space displacement Δx = (Δx i , Δyi , Δz i );

[0011] S4. Calculate the three-dimensional velocity field based on the object space displacement Δx and the time step Δt.

[0012] Furthermore, in step S1, the specific process includes the following steps:

[0013] S11. Define the object space displacement to be solved as Δx = (Δx i , Δy i , Δz i );

[0014] S12. Establish the second-order expansion of the apparent displacement relationship ΔX based on the second-order Taylor formula expansion, that is:

[0015]

[0016] where x = (x i , y i , z i ) is the object space coordinate; F is the projection function; F(x) = X, which is the projection relationship between the image plane coordinate X = (X i , Y i ) and the object space coordinate x = (x i , y i , z i ); are the derivatives of F(x) in the x, y, and z directions; Δx T is the transpose of the object space displacement Δx; H is the Hessian matrix of the projection function F;

[0017] S13. Based on the second-order expansion of the displacement relationship ΔX, determine the expansions of the four image plane components ΔX L , ΔY L , ΔX R , ΔY R , that is:

[0018]

[0019] where the image plane components ΔX L , ΔY L , ΔX R , ΔY R represent the pixel displacements on the left and right binocular images; Δx j , Δx k are formula compact representation symbols, representing the pairwise products of Δx i , Δy i , Δz i ;

[0020] S14. Combine the relationships of the four image plane components ΔX L , ΔY L , ΔX R , ΔY R to obtain a non - linear displacement model based on the relationship between the apparent two - dimensional displacement in the image plane and the true three - dimensional displacement in the object space.

[0021] Furthermore, the expression of the non - linear displacement model is:

[0022]

[0023] In the formula, F ij is the derivative of the i - th component of the projection function with respect to the coordinate component j; H i,jk represents the second - order partial derivative of the i - th projection component with respect to Δx j and Δx k .

[0024] Furthermore, in step S3, the specific process includes the following steps:

[0025] S31. Let the current estimated value be Δx (k) , and the iterative update amount be δx. Use the Newton - Raphson iteration method to determine the iterative formula as:

[0026] Δx (k+1) = Δx (k) + δx

[0027] where Δx (k+1) is the initial displacement value in the world coordinate system after the (k + 1) - th iteration;

[0028] S32. Obtain the iterative update amount δx by solving the linear equation system. The linear equation system is:

[0029] J(Δx (k) )δx = - Los(Δx (k) )

[0030] In the formula, J is the Jacobian matrix, representing the first - order derivative of the residual Loss with respect to Δx, that is:

[0031] S33. Use the Gauss - Newton iteration method to simplify and obtain the iterative equation and solve for the iterative update amount δx;

[0032] S34. Define the update strategy of the damping factor λ, that is:

[0033]

[0034] Wherein, λ0 is the initial damping factor; λ_new is the updated damping factor;

[0035] S35. Under the update strategy, a damping factor λ is introduced to ensure numerical stability in solving the iterative update amount δx, that is:

[0036]

[0037] Wherein, I is the identity matrix;

[0038] S36. Iteratively optimize and solve the residual function Loss(Δx) according to the iterative update amount δx after introducing the damping factor λ. If ||δx|| < the iterative threshold ∈ or the residual is small enough, stop the iteration.

[0039] Further, in step S33, the specific process includes:

[0040] Let:

[0041]

[0042] At this time, the iterative equation is:

[0043]

[0044] The solution of the iterative update amount δx is:

[0045]

[0046] Wherein, is the transpose of the Jacobian matrix J, where the derivatives of F(x) with respect to the three directions of x, y, and z form the Jacobian matrix J.

[0047] Further, the value of the iterative threshold ∈ is 1e -3 or 1e -6 .

[0048] Further, the calculation formula of the three-dimensional velocity field is:

[0049] v = Δx / Δt

[0050] Wherein, v represents the reconstructed velocity field.

[0051] Further, the out-of-plane velocity is w among the three velocity components u, v, and w in the two-dimensional three-component, which is the velocity perpendicular to the observation plane and is called the out-of-plane velocity.

[0052] By means of the above technical solution, the present invention provides a method for measuring the out-of-plane velocity of particle image velocimetry, which at least has the following beneficial effects:

[0053] 1. The present invention constructs a non - linear displacement model containing second - order Hessian terms to accurately describe the coupling effect between out - of - plane displacement and image distortion. In scenarios with large out - of - plane velocities or in the face of strong shear and vortex flow fields, it can ensure the stability of errors, thereby significantly improving the calculation accuracy of out - of - plane velocities.

[0054] 2. The present invention uses an iterative optimization algorithm to solve the non - linear displacement model to obtain the object - space displacement. At the same time, it combines a regularization strategy to ensure the stability of errors, thereby reducing errors and accelerating the iterative solution speed. BRIEF DESCRIPTION OF THE DRAWINGS

[0055] The drawings described herein are used to provide a further understanding of the present application and form a part of the present application. The illustrative embodiments of the present application and their descriptions are used to explain the present application and do not constitute an improper limitation of the present application. In the drawings:

[0056] Figure 1 It is a comparison result diagram of the out - of - plane displacement error in the present invention. DETAILED DESCRIPTION OF THE EMBODIMENTS

[0057] To make the above - mentioned objects, features, and advantages of the present invention more obvious and understandable, the present invention will be further described in detail below with reference to the drawings and specific embodiments. Thereby, a full understanding of how the present application uses technical means to solve technical problems and achieve technical effects can be obtained and implemented accordingly.

[0058] The traditional particle image velocimetry (PIV) technology analyzes the apparent displacement of a pair of particle images and uses first - order Taylor expansion to establish a linear displacement model to invert the three - dimensional velocity field, as shown in Equation (3) below. However, the first - order model ignores the high - order non - linear terms, resulting in the following problems:

[0059] 1. Significant out - of - plane velocity error: The out - of - plane displacement (Δz) will cause image scaling and distortion, and its non - linear effect cannot be accurately described by the linear model at large displacements.

[0060] 2. Poor adaptability to complex flows: In strong shear or vortex flow fields, the influence of high - order terms of the velocity gradient on the apparent displacement is exacerbated, and the accuracy of the linear model decreases.

[0061] The principle of using first - order Taylor expansion to establish a linear displacement model to invert the three - dimensional velocity field is described as follows:

[0062] Let the image - plane coordinates be X=(X i , Y i ), and the object - space coordinates be x=(x i , y i , z i ). Then the projection relationship between the two can be expressed as:

[0063] X = F(x) (1)

[0064] Accordingly, the relationship between the displacement in the image plane and the displacement in the object space is satisfied as follows:

[0065] ΔX = F(x + Δx) - F(x) (2)

[0066] The first-order expansion of Equation (2) is:

[0067]

[0068] where is called the imaging magnification matrix; Δx represents a very small offset (Δx, Δy, Δz) at the actual physical space point (x i , y i , z i ).

[0069] According to Equation (3), the apparent two-dimensional displacement in the image plane is:

[0070] X L = (ΔX L , ΔY L )

[0071] X R = (ΔX R , ΔY R )

[0072] where, (ΔX L , ΔY L ) and (ΔX R , ΔY R ) are the pixel displacements on the left and right binocular images.

[0073] Then, the relationship between the apparent two-dimensional displacement (X L , X R ) and the true three-dimensional displacement Δx = (Δx i , Δy i , Δz i ) in the object space is satisfied as follows:

[0074]

[0075] where, F ij is the derivative of the i-th component of the projection function with respect to the coordinate component j. For any point x in the flow field, according to the projection function F, it is mapped to two points X L and X R in the two image planes, and the apparent two-dimensional displacements ΔX L and ΔX R of the two points are calculated respectively according to the standard two-dimensional PIV, the true three-dimensional displacement Δx of point x can be reconstructed according to Equation (4).

[0076] Obviously, the key to solving the above reconstruction problem is the projection function F between the image plane and the object space. Although the theoretical derivation can be carried out based on the pinhole imaging model, the actual operation is difficult, and the theoretical model cannot consider problems such as lens distortion. Therefore, in practical applications, the projection relationship is usually approximated as a polynomial shown in Equation (5), and the unknown coefficients are obtained by the least squares method according to the calibration experiment.

[0077]

[0078] where, a0 to a 18 is the result of calibration. The object space coordinates x = (x i , y i , z i ) to the relationship of the image plane coordinates X = (X i , Y i ) described by the polynomial model is a commonly used method in the PIV field.

[0079] As described above, when establishing a linear displacement model through first-order Taylor expansion, the high-order terms are ignored, resulting in significant errors in the presence of large out-of-plane displacements or complex flows. Even by further increasing the calibration accuracy or optical compensation, the error loss caused by discarding the high-order terms in the principle itself cannot be solved. This embodiment proposes a method for measuring the out-of-plane velocity of particle image velocimetry, specifically a particle image velocimetry (PIV) method based on second-order Taylor expansion and non-linear optimization algorithm, which is used to improve the accuracy of the out-of-plane velocity in two-dimensional three-component (2D3C) measurement. Among them, the three components in the two-dimensional three-component refer to the three velocity components u, v, and w, where w is the velocity perpendicular to the observation plane and is called the out-of-plane velocity. By constructing a non-linear displacement model containing the second-order Hessian term in this embodiment, the calculation accuracy of the out-of-plane velocity can be significantly improved. The method includes the following steps:

[0080] S1. Based on the displacement relationship ΔX between the image plane and the object space, construct a non-linear displacement model containing the second-order Hessian term; in step S1, the specific process includes the following steps:

[0081] S11. Define the object space displacement Δx = (Δx i , Δy i , Δz i ) to be solved;

[0082] S12. Based on the second-order Taylor formula expansion, establish the second-order expansion of the apparent displacement relationship ΔX, that is:

[0083]

[0084] where \(x=(x i ,y i ,z i )\) is the object space coordinate; \(F\) is the projection function; \(F(x)=X\), where \(X=(X i ,Y i )\) is the projection relationship between the image plane coordinate and the object space coordinate \(x=(x i ,y i ,z i )\); is the derivative of \(F(x)\) in the \(x\), \(y\), and \(z\) directions; \(\Delta x T \) is the transpose of the object space displacement \(\Delta x\); \(H\) is the Hessian matrix (second derivative) of the projection function \(F\), obtained by numerical differentiation.

[0085] S13. Determine the expansions of the four image plane components \(\Delta X L ,\Delta Y L ,\Delta X R ,\Delta Y R \) based on the second-order expansion of the displacement relationship \(\Delta X\), i.e.:

[0086]

[0087] where the image plane components \(\Delta X L ,\Delta Y L ,\Delta X R ,\Delta Y R \) represent the pixel displacements on the left and right binocular images; \(\Delta x j \), \(\Delta x k \) are formula compact representation symbols, representing the pairwise products of \(\Delta x i ,\Delta y i ,\Delta z i \).

[0088] S14. Combine the relationships of the four image plane components \(\Delta X L ,\Delta Y L ,\Delta X R ,\Delta Y R \) to obtain a non-linear displacement model based on the relationship between the apparent two-dimensional displacement in the image plane and the true three-dimensional displacement in the object space. The expression is:

[0089]

[0090] where \(F ij \) is the derivative of the \(i\)-th component of the projection function \) with respect to the coordinate component \(j\); \(H i,jk \) represents the \(i\)-th projection component with respect to \(\Delta x j \) and \(\Delta x kThe second-order partial derivative. The nonlinear displacement model is a nonlinear system, and the physical space displacement Δx = (Δx i , Δy i , Δz i ) needs to be solved.

[0091] S2. Establish a residual function Loss(Δx) and an optimization objective function E(Δx) for optimizing the nonlinear displacement model. Among them, the residual function Loss(Δx) represents the residual between the observed displacement and the predicted displacement, and the expression is:

[0092]

[0093] The goal of the optimization objective function is to minimize the sum of the squares of the residuals, that is:

[0094]

[0095] Among them, E(Δx) represents minimizing the sum of the squares of the residuals.

[0096] S3. Iteratively optimize and solve the residual function Loss(Δx) with the optimization objective function E(Δx) as the goal to obtain the physical space displacement Δx = (Δx i , Δy i , Δz i ); In step S3, the specific process includes the following steps:

[0097] S31. Let the current estimated value be Δx (k) and the iterative update amount be δx. Use the Newton-Raphson iteration method to determine the iterative formula as:

[0098] Δx (k+1) = Δx (k) + δx

[0099] Among them, Δx (k+1) is the initial displacement value in the world coordinate system after the (k + 1)-th iteration.

[0100] S32. Obtain the iterative update amount δx by solving a system of linear equations. The system of linear equations is:

[0101] J(Δx (k) )δx = -Loss(Δx (k) )

[0102] In the formula, J is the Jacobian matrix, which represents the first-order derivative of the residual Loss with respect to Δx, that is:

[0103] S33. Use the Gauss - Newton iteration method to simplify and obtain the iteration equation, and solve for the iteration update amount δx. For large - scale problems, the second - order Hessian term can be ignored, and only the first - order Jacobian approximation is used for iteration. Let:

[0104]

[0105] At this time, the iteration equation is:

[0106]

[0107] The solution of the iteration update amount δx is:

[0108]

[0109] In the formula, is the transpose of the Jacobian matrix J, where the derivatives of F(x) with respect to the x, y, and z directions form the Jacobian matrix J.

[0110] Here, it needs to be further explained that the purpose of ignoring the high - order terms here is to speed up the computational amount of a single iteration. Different from the Taylor expansion of the formula (2) of the traditional first - order linear model that ignores the high - order terms to obtain the formula (3), this will not cause accuracy loss, but only may cause an increase in the number of iterations (because δx is approximately calculated with accuracy loss, so the speed of approaching the correct result in each iteration may become smaller). However, since a large amount of calculation is saved in a single iteration, overall, the Gauss - Newton method is still 3 - 5 times faster than the Newton - Raphson method.

[0111] S34. Let the initial damping factor be λ0, and define the update strategy of the damping factor λ, that is:

[0112]

[0113] In the formula, λ0 = 1e -3 ; λ_new is the updated damping factor.

[0114] S35. Under the update strategy, introduce the damping factor λ used to ensure numerical stability in solving the iteration update amount δx, that is:

[0115]

[0116] In the formula, I is the identity matrix, and the dimension is the same as consistent.

[0117] S36. Iteratively optimize and solve the residual function Loss(Δx) according to the iterative update amount δx after introducing the damping factor λ. If ||δx|| < the iterative threshold ∈ or the residual is small enough, stop the iteration. Generally, a small enough residual means that the threshold ∈ needs to reach 1e -6 , of course, 1e can be selected according to the actual iterative precision requirements -3 , which can reduce the number of iterations to a certain extent and converge faster, but the result is necessarily not accurate. In this embodiment, 1e is used for the sake of precision -6 .

[0118] As shown in Table 1, it is the algorithm flow of the Gauss - Newton iterative method. By adding the damping factor λ of the Levenberg - Marquardt regularization term in the simplification process, the numerical stability is improved

[0119] Table 1 Algorithm flow of the Gauss - Newton iterative method

[0120]

[0121] In this embodiment, regularization processing is performed on the solution of the iterative update amount δx. If is singular (i.e., the determinant is 0), it is no longer invertible at this time. If the residual function Loss(Δx) is too large, for a non - linear model, the Gauss - Newton iteration may diverge. Therefore, in order for the iteration to proceed normally, the damping factor λ is added in combination with the regularization processing (Levenberg - Marquardt) to ensure the numerical stability

[0122] Specifically, the present invention uses an iterative optimization and solution algorithm to solve the object - space displacement of the non - linear displacement model, and at the same time combines a regularization strategy to ensure the stability of the error, thereby reducing the error and accelerating the iterative solution speed

[0123] S4. According to the object - space displacement Δx, calculate the three - dimensional velocity field in combination with the time step Δt. The calculation formula is

[0124] v = Δx / Δt

[0125] In the formula, v represents the reconstructed velocity field

[0126] Compared with the existing technology, in this embodiment, through experimental simulation and comparison with the first - order linear model, the error results are as Figure 1 shown. Inject known out - of - plane displacements (Δz varies from 0 - 20 pixels) into the synthetic flow field. The average error of the traditional first - order linear model is 12%, and the error increases with the change of the out - of - plane displacement Δz. While the method proposed in this embodiment reduces the average error to 3%, and the error is stable

[0127] Specifically, the present invention constructs a non-linear displacement model containing second-order Hessian terms to accurately describe the coupling effect between out-of-plane displacement and image distortion. In scenarios with large out-of-plane velocities or in the face of strong shear and vortex flow fields, it can ensure the stability of errors, thereby significantly improving the calculation accuracy of out-of-plane velocities.

[0128] Those of ordinary skill in the art can understand that all or part of the steps in the methods of the above embodiments can be completed by instructing relevant hardware through a program. Therefore, the present application can adopt the form of a complete hardware embodiment, a complete software embodiment, or an embodiment combining software and hardware aspects. Moreover, the present application can adopt the form of a computer program product implemented on one or more computer-usable storage media (including but not limited to disk memories, CD-ROMs, optical memories, etc.) containing computer-usable program codes.

[0129] The above embodiments have introduced the present invention in detail. Specific examples are used in this article to elaborate on the principles and implementation methods of the present invention. The description of the above embodiments is only used to help understand the method and its core idea of the present invention; at the same time, for those of ordinary skill in the art, according to the idea of the present invention, there will be changes in the specific implementation methods and application scopes. In summary, the content of this specification should not be construed as a limitation on the present invention.

Claims

1. An out-of-plane velocity measurement method for particle image velocimetry, characterized in that The method includes the following steps: S1. Construct a non - linear displacement model containing second - order Hessian terms based on the displacement relationship ΔX between the image plane and the object space; S2. Establish a residual function Loss(Δx) and an optimization objective function E(Δx) for optimizing the non - linear displacement model; S3. Iteratively optimize and solve the residual function Loss(Δx) with the goal of optimizing the objective function E(Δx) to obtain the object space displacement Δx = (Δx i , Δy i , Δz i ); S4. Calculate the three - dimensional velocity field according to the object - space displacement Δx and in combination with the time step Δt.

2. The out-of-plane velocity measurement method according to claim 1, characterized in that In step S1, the specific process includes the following steps: S11. Define the object space displacement to be solved as Δx=(Δx i , Δy i , Δz i ); S12. Establish a second - order expansion of the apparent displacement relationship ΔX based on the second - order Taylor formula expansion, that is: where \(x=(x i ,y i ,z i )\) is the object space coordinate; \(F\) is the projection function; \(F(x)=X\) is the projection relationship between the image plane coordinate \(X=(X i ,Y i )\) and the object space coordinate \(x=(x i ,y i ,z i )\); are the derivatives of \(F(x)\) in the \(x\), \(y\), and \(z\) directions; \(\Delta x T \) is the transpose of the object space displacement \(\Delta x\); \(H\) is the Hessian matrix of the projection function \(F\); S13. Determine the expansions of the four image plane components ΔX L , ΔY L , ΔX R , ΔY R based on the second-order expansion of the displacement relationship ΔX, that is: In the formula, the image plane components ΔX L , ΔY L , ΔX R , ΔY R represent the pixel displacements on the left and right binocular images; Δx j , Δx k are formula compact representation symbols, representing the pairwise products of Δx i , Δy i , Δz i ; S14. Combine the relationships of the four image plane components ΔX L , ΔY L , ΔX R , ΔY R to obtain a non-linear displacement model based on the relationship between the apparent two-dimensional displacement in the image plane and the true three-dimensional displacement in the object space.

3. The out-of-plane velocity measurement method according to claim 2, wherein The expression of the non - linear displacement model is: where F ij is the derivative of the i-th component of the projection function with respect to the coordinate component j; H i,jk represents the second-order partial derivative of the i-th projection component with respect to Δx j and Δx k .

4. The out-of-plane velocity measurement method according to claim 3, characterized in that In step S3, the specific process includes the following steps: S31. Let the current estimated value be Δx (k) . Let the iterative update amount be δx, and the iterative formula is determined by the Newton-Raphson method as follows: Δx (k+1) = Δx (k) + δx where, Δx (k+1) is the initial displacement value in the world coordinate system after the (k + 1)-th iteration; S32. Obtain the iterative update quantity δx by solving a linear equation system, and the linear equation system is: J(Δx (k) )δx = -Loss(Δx (k) ) where J is the Jacobian matrix, representing the first-order derivative of the residual Loss with respect to Δx, that is: S33. Simplify using the Gauss - Newton iteration method to obtain an iterative equation and solve for the iterative update quantity δx; S34. Define the update strategy of the damping factor λ, that is: In the formula, λ0 is the initial damping factor; λ_new is the updated damping factor; S35. Under the update strategy, introduce the damping factor λ used to ensure numerical stability in solving the iterative update quantity δx, that is: In the formula, I is the identity matrix; S36. Iteratively optimize and solve the residual function Loss(Δx) according to the iterative update quantity δx after introducing the damping factor λ. If ||δx|| < iterative threshold ∈ or the residual is small enough, stop the iteration.

5. The out-of-plane velocity measurement method according to claim 4, wherein In step S33, the specific process includes: Let: At this time, the iterative equation is: The solution of the iterative update quantity δx is: In the formula, is the transpose of the Jacobian matrix J, where the derivatives of F(x) with respect to the three directions of x, y, and z form the Jacobian matrix J.

6. The out-of-plane velocity measurement method according to claim 4, characterized in that The value of the iterative threshold ∈ is 1e -3 or 1e -6 .

7. The out-of-plane velocity measurement method according to claim 1, wherein The calculation formula of the three - dimensional velocity field is: v = Δx / Δt In the formula, v represents the reconstructed velocity field.

8. The out-of-plane velocity measurement method according to any one of claims 1-7, characterized in that, The out - of - plane velocity is w among the three velocity components u, v, and w in the two - dimensional three - component, which is the velocity perpendicular to the observation plane and is called the out - of - plane velocity.

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