A method for measuring vertical flow velocity of incompressible fluid

Abstract

The present invention discloses a method for measuring the vertical velocity of an incompressible fluid. First, the height of the entire flow field is layered to obtain the observation data of the horizontal velocity of each layer. Secondly, a horizontal analysis field is constructed by numerical differentiation method. Finally, the vertical velocity on each layer is obtained by variational optimal analysis method to construct the vertical flow field of the entire fluid. The present invention abstracts the mesoscale ocean vortex, and for the horizontal observation flow field data, the Gignoff regularization method is used to construct the horizontal analysis flow field, thereby avoiding the influence of observation error and low resolution of observation data on the calculation of vertical velocity. Finally, the variational optimal analysis method is used to correct the calculated vertical velocity, thereby providing a feasible method for calculating the vertical velocity of meteorological and oceanic phenomena such as typhoons and mesoscale vortices.

Description

Technical Field

[0001] The invention relates to a method for measuring the vertical flow velocity of an incompressible fluid. Background Art

[0002] Vertical velocity plays a very important role in the diagnosis and forecast of atmospheric and oceanic motion. Since the vertical velocity in large-scale atmospheric and oceanic motion is very small, only about one percent and one thousandth of the horizontal velocity, it is difficult to directly observe it with existing observation instruments and must be diagnosed by other factors. For incompressible fluids, the continuity equation is:

[0003]

[0004] Among them, x, y, z are the coordinate axis directions, u, v, w are the velocity components corresponding to the coordinate axis directions, when the upper and lower boundary conditions of the horizontal velocity components u, v and the vertical velocity component w are known, the variational optimal analysis method can be used to deal with it. However, this method involves the problem of deriving u and v. Since u and v are obtained by observation, errors are inevitable. On the other hand, u and v are discrete point values, which will cause the following problems:

[0005] When the observed values ​​of u,v When there is a random error, The derivative operation will produce a large error, so the finite difference method is used to perform the derivative. This causes a large error, and thus the further error in calculating the vertical velocity is also large.

[0006] because It is the value at the observation point. The grid is very discontinuous. How to interpolate the value at the grid point to improve the resolution?

[0007] For the ocean, the vertical flow field on the sea surface is difficult to obtain accurately, so the finite difference method needs to be corrected. Summary of the invention

[0008] Purpose of the invention: In view of the above-mentioned prior art, a method for measuring the vertical flow velocity of an incompressible fluid is proposed. The horizontal analysis field is reconstructed through the horizontal observation data of the large-scale flow field, so that the problem of numerical differential ill-posedness can be overcome in the process of solving the divergence. On this basis, the vertical velocity of the fluid is solved by the variational optimization method.

[0009] Technical solution: A method for measuring the vertical flow velocity of an incompressible fluid, comprising: first, stratifying the incompressible fluid, and obtaining the horizontal flow velocity on each layer through observation; then, obtaining a continuous horizontal analysis field from the discrete observation data of the horizontal flow velocity through the Tikhonov regularization method, and calculating the partial derivative of the horizontal analysis field, thereby obtaining its divergence; secondly, constructing a variational optimization function using a variational optimal analysis method, taking the continuity equation of the incompressible fluid as a constraint condition, and obtaining the vertical flow velocity.

[0010] Beneficial effects: (1) By variation, the symmetry center and the maximum symmetric vortex of the ocean vortex are obtained, making them an independent part of the incompressible fluid, thereby satisfying the ideal equation of the incompressible fluid.

[0011] (2) Through the Tikhonov regularization method, the discrete horizontal observation data are interpolated to obtain a continuous horizontal analysis field, which overcomes the increase in calculation error caused by the increase in the resolution of the observation data, thereby improving the resolution. At the same time, the calculation error can also be effectively controlled when the observation error is unknown.

[0012] (3) For the incompressible fluid bottom layer, w0 = 0 can be set, and the boundary condition of the top layer can be obtained by It can be calculated. BRIEF DESCRIPTION OF THE DRAWINGS

[0013] Figure 1 It is a schematic diagram of the model description steps of the present invention;

[0014] Figure 2 It is the specific calculation steps of using Tikhonov regularization method to obtain the horizontal analysis field. DETAILED DESCRIPTION

[0015] The present invention will be further explained below in conjunction with the accompanying drawings.

[0016] like Figure 1 As shown in the figure, a method for measuring the vertical velocity of an incompressible fluid is to first abstract the typhoon, mesoscale vortex, etc. into an incompressible whole, thereby satisfying the continuity equation of the incompressible fluid. The entire fluid is layered, and the horizontal velocity data of each layer is obtained through observation; secondly, the horizontal velocity data of each layer is subjected to the Tikhonov regularization method to obtain a continuous horizontal analysis field, and the partial derivative of the horizontal analysis field is calculated to obtain the divergence of the horizontal analysis field; finally, the continuity equation of the incompressible fluid is solved The boundary conditions are respectively obtained by inverting the sea surface velocity from satellite data and assuming that the bottom velocity is 0, and the vertical velocity on each layer is obtained by solving it. The variational optimal analysis method is used in the solution process to control the error.

[0017] In the above process, the key step is to obtain the horizontal analysis field through the observation data of horizontal flow velocity. Figure 2 Describes the process of this step, combined with Figure 2 Give a detailed explanation. Let y = y(x) be a function defined on [0,1], n be a natural number, Δ = {0 = x0 < x1 < ... < x n =1} is a partition on [0,1], let δ be a given constant used to represent the observation error of the observed data.

[0018] Remember h i =x i+1 -x i ,i=1,2,…,n-1,

[0019] Given a function at point x i Observed value at satisfy:

[0020]

[0021] in, For point x i Observational data on

[0022] We hope to find the function f * (x), abbreviated as f * , so that f * '(x) is an approximation of the function y'(x).

[0023] Without loss of generality, let That is, the observations at the two end points are exact. Define the functional:

[0024]

[0025] Here α is the regularization parameter, For point x j The observation data on Indicates L 2 Norm. The problem is further transformed into:

[0026] 1: Find the function f * ∈H 2 (0,1), satisfying f * (0) = y(0), f * (1) = y(1), so that J(f * )=minJ, where H 2 (0,1) represents the class of functions that are Lebesgure-differentiable of order 2 on (0,1).

[0027] 2: If f* There is a way to choose the regularization parameter α related to δ so that f * ′(x) is an approximation of y′(x)?

[0028] Theoretically, it can be proved that for any α>0, the solution f of problem 1 is * exists and is unique, and f * is a cubic natural spline function that partitions a piece on Δ; let α = δ 2 When , the error estimate is as follows:

[0029]

[0030] When h→0, δ→0, then f * ′→y′. It can be seen that the regularization parameter α=δ 2 , the solution to Problem 1 converges.

[0031] f * (x) is a piecewise cubic natural spline function, in the subinterval [x j ,x j+1 The expression on ] is:

[0032] f * (x) = a j +b j (xx j )+c j (xx j ) 2 +d j (xx j ) 3 ,j=0,1,…,n-1;

[0033] Among them, a j 、b j 、c j d j is the coefficient;

[0034] Therefore, there are a total of 4n unknown coefficients, which are obtained by solving the following system of equations, and the solution is unique:

[0035] f * (i) (x j +)-f * (i) (x j -)=0, i=0,1,2; j=1,2,...,n-1;

[0036] where f * (j) Represents the function f *The j-order derivative of , let T and Q be the following (n-1)×(n-1) order matrices:

[0037]

[0038] a=(a1,a2,…,a n-1 ) T ,c=(c1,c2,…,c n-1 ) T ,

[0039] Therefore c0=0,d n-1 =-c n-1 / (3h),d j =(c j+1 -c j ) / (3h),j=0,1,…,n-1

[0040]

[0041] b j =(a j+1 -a j ) / hc j hd j h 2 ,j=0,1,…,n-1;

[0042] You can reconstruct the function f * And find f * ′.

[0043] Based on the above method, the observed horizontal flow field is assumed to be and The vertical observation component is unknown. Consider the region Ω×[0,H] and find the modified flow field (u,v,w) that satisfies:

[0044]

[0045] Constraints:

[0046]

[0047] Let w| z＝0 =w0,w| z＝H =w H , introducing the Largrange multiplier λ(x,y), we have:

[0048]

[0049] Using variational calculus, we get:

[0050]

[0051] According to Green's formula:

[0052]

[0053] is the normal vector of Ω. According to the arbitrariness of δu, δv, δw, we get:

[0054]

[0055] λ satisfies:

[0056] Here, the divergence Obtained by numerical differentiation;

[0057] The relaxation method can be used to obtain λ(x,y), and then u and v can be obtained, so that w is:

[0058]

[0059] The above is only a preferred embodiment of the present invention. It should be pointed out that for ordinary technicians in this technical field, several improvements and modifications can be made without departing from the principle of the present invention. These improvements and modifications should also be regarded as the scope of protection of the present invention.

Claims

1. A method for measuring the vertical flow velocity of an incompressible fluid, characterized in that: include: Firstly, the incompressible fluid is layered, and the horizontal velocity on each layer is obtained by observation. Then, the continuous horizontal analysis field is obtained from the discrete observation data of the horizontal velocity through the Tikhonov regularization method, and the partial derivative of the horizontal analysis field is calculated to obtain its divergence. Secondly, the variational optimal analysis method is used to construct the variational optimization function, and the continuity equation of the incompressible fluid is used as a constraint to obtain the vertical flow velocity; In the xyz coordinate system, the discrete observation data of the horizontal flow field include and The corresponding continuous horizontal analysis fields are u(x, y, z) and v(x, y, z). Find the vertical flow field w(x, y, z) in the region Ω×[0, H] so that the modified flow field (u, v, w) satisfies: Where, Ω represents the fluid projection area, and H represents the height; The constraints are: The boundary conditions are: In z＝0 =w0,w z＝H =in H w0 represents the vertical flow velocity at the bottom, w H Indicates the vertical flow velocity at the top; Using the Lagrange multiplier method and the variational optimal analysis method, we get: Where λ is the Lagrange multiplier, satisfying: Among them, the divergence Use the relaxation method to obtain λ(x,y), and further find u,v, so that w is:

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