PREDICTION METHOD FOR MAXIMUM VELOCITY PROFILE IN WAVE BOUNDARY LAYER BASED ON VELOCITY DEFECT FUNCTIONS

Abstract

The present invention discloses a prediction method for a maximum velocity profile in a wave boundary layer based on velocity defect functions. The method overcomes the theoretical defects of the existing velocity defect functions. That is, the velocity profile in a turbulent wave boundary layer cannot be realized; and in addition, without the assumption of linear wave conditions, the method is suitable for nonlinear waves and at the same time, for a small A/k, range, and can be extended to the analysis and prediction for the maximum velocity profile under the condition that the spatial distribution of roughness elements of gravel seabed, etc. obviously affects the flow structure of the boundary layer. The present invention can be directly applied to the analysis and prediction for physical quantities/processes, such as characteristics of the wave boundary layer, stress of underwater structures, and starting and transport of submarine sediments.

Claims

1. A prediction method for a maximum velocity profile in a wave boundary layer based on velocity defect functions, comprising the following steps: A. establishing a prediction formula of the maximum velocity profile in the wave boundary layer by considering the effects of the seabed roughness height and the boundary layer thickness on velocity distribution, introducing the length scale into the velocity defect functions, to obtain the three-parameter velocity defect function χ(z) for λ.sub.1, λ.sub.2 and p; $\begin{array}{cc}\chi \left(z\right)={\exp \left(-\frac{z}{{\lambda }_1}\right)}^p{\cos \left(-\frac{z}{{\lambda }_2}\right)}^p& \left(1\right)\end{array}$ $\begin{array}{cc}u\left(z\right)=\left[1-\chi \left(z\right)\right]U_m& \left(2\right)\end{array}$ where λ.sub.1 and λ.sub.2 are length scales which are respectively used for describing the effects of the seabed roughness height and the boundary layer thickness on the velocity distribution; p is an exponential parameter which is used for adjusting a condition under which the velocity distribution doesn't meet the logarithmic rate law; u(z) indicates the value of the maximum horizontal velocity at the vertical coordinate z; and U.sub.m indicates the velocity amplitude of wave water quality point motion in a free-flowing region outside the boundary layer; the maximum flow velocity profile under a wave condition is indicated by vertical coordinates η.sub.1, η.sub.2 and δ.sub.J; the velocity overshoot region exits inside the wave boundary layer, and η.sub.1 and η.sub.2 respectively indicate a lower boundary and an upper boundary of the velocity overshoot region, that is, η.sub.2−η.sub.1 indicates a scope of the velocity overshoot region; δ.sub.J indicates a vertical coordinate corresponding to the maximum velocity in the velocity overshoot region; according to definitions of vertical coordinate physical quantities η.sub.1, η.sub.2 and δ.sub.J, binding conditions are obtained: u(η.sub.1)=U.sub.m; u(η.sub.2)=U.sub.m; when z=δ.sub.1, ∂.sub.χ/∂z=0; and by combining these constraints with formula (1), the expressions of η.sub.1, η.sub.2 and δ.sub.J are obtained; $\begin{array}{cc}{\eta }_1={\left(\frac{\pi }{2}\right)}^{1/p}{\lambda }_2,{\eta }_2={\left(\frac{3\pi }{2}\right)}^{1/p}{\lambda }_2& \left(3\right)\end{array}$ $\begin{array}{cc}{\delta }_J=\left[\pi -\mathrm{acr}\tan \left(\frac{{\lambda }_2^p}{{\lambda }_1^p}\right)\right]{\lambda }_2& \left(4\right)\end{array}$ the maximum velocity deviation function ξ.sub.max is obtained based on formulae (1), (3) and (4): $\begin{array}{cc}{\xi }_{\max }=-{\chi }_r\left({\delta }_J\right)=\frac{{\lambda }_1^p}{\sqrt{{\lambda }_1^{2p}+{\lambda }_2^{2p}}}\exp \left[-\frac{{\lambda }_2^p}{{\lambda }_1^p}\left(\pi -\mathrm{acr}\tan \left(\frac{{\lambda }_2^p}{{\lambda }_1^p}\right)\right)\right]& \left(5\right)\end{array}$ B. determining length scales λ.sub.1, λ.sub.1, and the exponential parameter p by analyzing the formula (5) and physical experimental results, determining the length scales λ.sub.1, λ.sub.2 and the exponential parameter p in formula (1) using a least square-fitting method; a coefficient prediction formula: $\begin{array}{cc}\frac{{\lambda }_1}{k_s}=0.06{\left(\frac{A}{k_s}\right)}^{0.5}+0\text{.041}& \left(6\right)\end{array}$ $\begin{array}{cc}\frac{{\lambda }_2}{k_s}=0.08{\left(\frac{A}{k_s}\right)}^{0.48}& \left(7\right)\end{array}$ $\begin{array}{cc}p=1.154-0.142\lg \left(\frac{A}{k_s}\right)& \left(8\right)\end{array}$ where k.sub.s indicates seabed roughness, and 2.5 times of the element characteristic diameter of the seabed roughness is taken; A indicates displacement amplitude of the water quality point motion of the wave outside the boundary layer, which is calculated by a wave theory; and in the case of nonlinear wave, the maximum displacement amplitude of the water quality point motion of the wave is taken; and substituting formulae (6), (7) and (8) into formulae (1) and (2) to achieve the prediction for the maximum velocity profile in the wave boundary layer.

2. A prediction method for a maximum velocity profile in a wave boundary layer based on velocity defect functions according to claim 1, in the maximum velocity deviation function ξ.sub.max, χ(z) degrades to a two-parameter model when λ.sub.1=λ.sub.2=λ; and δ.sub.J=λ(3π/4).sup.1/p), ξ.sub.max=6.7%.

Description

DESCRIPTION OF DRAWINGS

[0020] FIG. 1 is a setup diagram for a physics experiment.

[0021] FIG. 2 shows a relationship between λ.sub.1/k.sub.s and A/k.sub.s, and a dotted line in the figure is fitting results of formula (9);

[0022] FIG. 3 shows a relationship between λ.sub.2/k.sub.s and A/k.sub.s, and a dotted line in the figure is fitting results of formula (10);

[0023] FIG. 4 shows a relationship between the exponent p and A/k.sub.s, and a dotted line in the figure is fitting results of formula (10); and

[0024] FIG. 5 shows comparison between the predicted value of a maximum velocity profile of a wave boundary layer and the results of physical experiments carried out by the present invention and others. The solid line in the figure is predication results using formula (4). FIG. 5(a)-FIG. 5(d) are respectively experimental results carried out by others, respectively Jonsson etc.(1976) case 02, Jensen etc.(1989) case 10, Dixen etc. (2008) case p4 and Vander etc. (2011) case S757012; FIG. 5(e)-FIG. 5(h) respectively show results of four working conditions in this method, wherein in a working condition 1, a wave period T=2.25s, U.sub.m=0.45 m/s, and A/k.sub.s=15.07; in a working condition 2, a wave period T=2.25s, U.sub.m=0.45 m/s, and A/k.sub.s=4.26; in a working condition 3, a wave period T=2.25s, U.sub.m=0.37 m/s, and A/k.sub.s=1.69; and in a working condition 4, a wave period T=2.25s, U.sub.m=0.37 m/s, and A/k.sub.s=1.44.

[0025] FIG. 6 is comparison between the predicted value of a maximum velocity overshoot max of a wave boundary layer and the results of physical experiments carried out by the present invention and others. The solid line in the figure is predicated results using formula (8). The current experimental working conditions 01 to 04 in FIG. 6 are consistent with the working conditions 1 to 4 in FIG. 5.

[0026] In the figures, 1—wave height meter; 2—ADV; 3—rough bottom bed; 4—transition slope; 5—wave making band; and 6—wave eliminating band.

DETAILED DESCRIPTION

[0027] The present invention is further illustrated below in combination with the drawings.

[0028] As shown in FIG. 1, the physical experiments conducted by adopting the method in the present invention are as follows:

[0029] the physical experiments involved in the present invention are carried out in an oil spilling tank of the State Key Laboratory of Coastal and Offshore Engineering, Dalian University of Technology. The tank is 23 m long, 0.8 m wide and 0.8 m deep. One end of the tank is equipped with a pusher plate type wave maker, to generate waves with a cycle range of 1.0s to 2.5s in the wave making band 5. The other end of the tank is installed with a slope type wave eliminating net for a wave eliminating band 6, to eliminate reflected waves. The test section is arranged in the middle of the tank, and a wave height meter 1 is arranged in the middle of the water surface; and the relevant physical experiment settings are shown in FIG. 1. The experimental terrain is made of concrete, which is 10 m long, 0.8 m wide and 0.13 m high. The transition slope 4 is set at the ratio of 1:15 on both ends of the terrain, to ensure that incident waves propagate slowly to the experimental terrain, and the experimental water depth is 0.4 m. An organic glass plate with length of 6 m and width of 0.8 m is laid on the concrete terrain for arranging a seabed model with different roughness heights. The definition of an experimental coordinate system is shown in FIG. 1, wherein, a horizontal direction is defined as an x-axis, and a travel direction of an incident wave is a positive x-axis; and the water depth direction is defined as a z-axis, a zero point thereof is located at a zero point of a theoretical bottom bed, and the direction of the water bottom pointing to the water surface is the positive z-axis.

[0030] In order to study the effect of seabed roughness k.sub.s on the characteristics of the wave boundary layer, four kinds of rough bottom beds 3 are set in the experiment, which are composed of quartz sand with median diameter d.sub.50=3.0 mm, glass spheres with average diameters D=10.6 mm and 26.7 mm, and irregular gravel respectively. The quartz sand and glass sphere rules are pasted on a smooth organic glass plate, and the gravel is directly tiled on the organic glass plate. In the experiment, acoustic doppler velocimetrys (ADV) are used for measuring the vertical distribution of the horizontal flow velocity. The spatial resolution of an ADV 2 is 1 mm, and the flow velocity of 35 measuring points are collected synchronically within the range of 3.5 cm. The distance from an ADV probe to the bottom bed is 7.5 cm, and a starting position through ADV measurement is at 4 cm below the probe; for the seabed composed of quartz sand, the horizontal flow velocity is considered to be homogeneous in a width direction of the tank because the corresponding seabed roughness height is small. Therefore, only one flow velocity measuring point is arranged in the middle of a central axis of an experimental tank; and however, for the seabed composed of glass spheres and gravel, the shape of roughness elements will have an obvious effect on the flow in the wave boundary layer, so it is necessary to arrange multiple measuring points and adopt an ensemble average data processing method, to obtain the average horizontal flow velocity distribution. The relevant calculation formulae are as follows:

[00008]$\begin{array}{cc}{\overline{u}}_i\left(\omega t,z\right)=\frac{1}{M}\underset{i=1}{\overset{M}{.Math.}}{u}_i\left[\omega \left(t+\left(i-1\right)T\right),z\right]& \left(12\right)\end{array}$$\begin{array}{cc}.Math.\overline{u}.Math.\left(\omega t,z\right)=\frac{1}{S}\underset{S}{\int }\overline{u}\left(\omega t,z\right)\mathrm{dS}=\left[\underset{j=1}{\overset{N}{.Math.}}{\overline{u}}_j\left(\omega t,z\right)\right]/N& \left(13\right)\end{array}$

where ū.sub.i indicates the average horizontal flow velocity during the period of the ith flow velocity measuring point at the coordinate z; M indicates the number of wave cycles, and in the process of data processing, M is greater than 30; w indicates the wave circle frequency, T indicates the wave period; <ū> indicates the average horizontal velocity after ū passes through space; S indicates an area of a flow velocity measuring region; and N indicates the number of flow velocity measuring points arranged in the experiment.

[0031] Two nonlinear second-order Stokes waves, named as w.sub.a and w.sub.b, are set up in the experiment, wherein, w.sub.a interacts with the rough seabed composed of quartz sand and glass spheres; and w.sub.b interacts with the gravel seabed. The present invention mainly focuses on a maximum velocity deviation function ξ.sub.m and the vertical distribution characteristics of maximum horizontal velocity, which are related to the maximum velocity amplitude U.sub.m and maximum displacement amplitude A of wave water quality point motion. The peak and trough of the second-order Stokes wave have asymmetrical distribution relative to a static water surface. Therefore, before formal physical experiments are carried out, a wave propagation experiment under the condition of a smooth bottom bed is firstly carried out in order to determine the basic parameters of wave propagation. In this part of the experiment, a time history line of the flow velocity at z=3 cm above the smooth bottom bed is measured by the ADV 2, and the measured results are taken as the free flow velocity unaffected by the boundary layer. Basic parameters of waves measured through the experiment are shown in Table 1:

TABLE-US-00001 TABLE 1 Basic Parameters of Nonlinear Second-order Stokes Wave Used in an Experiment No. T (s) U.sub.p (m/s) U.sub.p/U.sub.n A.sub.p (m) A.sub.p/A.sub.n w.sub.a 2.25 0.45 1.43 0.113 0.87 w.sub.b 2.25 0.37 1.54 0.0853 0.79
where U.sub.p and U.sub.n indicate amplitudes of horizontal flow velocity in the first half cycle and the second half cycle respectively; and A.sub.p and A.sub.n are horizontal motion amplitudes of wave water quality points in the first half cycle and the second half cycle respectively. As can be seen from Table 1, for the nonlinear waves adopted by the present invention, U.sub.p>U.sub.n and A.sub.p>A.sub.n. In order to obtain a maximum flow velocity profile in the wave boundary layer, U.sub.m=U.sub.p and A=A.sub.p are adopted in the subsequent analysis.

[0032] Comparative analyses of the present invention and physical experiments: in the experiment, the velocity in the wave boundary layer under different seabed roughness conditions is measured in real time by the ADV 2. The distribution of the maximum velocity profile in the boundary layer along the water depth is obtained through the analysis of formulae (12) and (13) and is compared with the prediction results of formulae (4) and (5). The relevant results are shown in FIG. 5. In order to further verify the validity of the analytical predication method proposed in the present invention, the relevant predication results are compared with the experimental results of others. As can be seen from FIG. 5, the prediction method for the maximum velocity profile in the wave boundary layer proposed in the present invention has high prediction accuracy, and the error between the predicated value and the measured value of the maximum velocity overshoot ξ.sub.max is less than 2%.

[0033] Through the analysis of the relevant data of the physical experiments carried out by the present invention and others, the relationship between two length parameters λ.sub.1, λ.sub.2 and the exponential parameter p in the velocity defect function proposed by the present invention and A/k.sub.s is analyzed in combination with formulae (6) and (7). The quantitative relationship between λ.sub.1/k.sub.s, λ.sub.2/k.sub.s and the exponential parameter p and A/k.sub.s is established, as shown in formulae (9)-(11) and FIGS. 2-4. The comparison between the maximum velocity overshoot Amax predicted by the prediction method of the present invention with the analytical data of the physical experiments conducted by the present invention and others is given in FIG. 6. As can be seen from the figure, the analysis results of the prediction method proposed by the present invention are in good agreement with the experiment results, which once again prove the validity of the prediction method of the maximum velocity profile of the wave boundary layer based on the velocity defect function proposed by the present invention.