CN103629120B - A kind of dynamic performance defining method being applicable to the smooth choma of multistage centrifugal pump - Google Patents

Abstract

The invention provides a kind of dynamic performance defining method being applicable to the smooth choma of high pressure multiple-stage centrifugal pump, described dynamic performance defining method is based on short sealing theory and have limit for length theoretical, described dynamic performance defining method comprises the steps: step one, utilize dimensional measuring instrument to determine the structural parameter of required high pressure multiple-stage centrifugal pump impeller port ring, and determine the operation operating mode of centrifugal pump; Step 2, sets up the governing equation group that equation of continuity, circumferential momentum equation and axial momentum equation form; Step 3, carries out dimensionless simplification to above-mentioned governing equation group and utilizes perturbation method to ask for full scale equation group perturbing about the single order of Perturbation and zeroth order the difference set of equation of form; Step 4, utilizes shooting method to solve dimensionless difference set of equation after simplification, and finally determines the size of the dynamic coefficients of high pressure multiple-stage centrifugal pump choma.The present invention is used for the determination of high pressure multiple-stage centrifugal pump choma dynamic performance, and Applicable scope is wider, calculates acquired results more accurate.

Description

A kind of dynamic performance defining method being applicable to the smooth choma of multistage centrifugal pump

Technical field

The present invention relates to dynamic performance and determine field, the present invention relates more specifically to a kind of dynamic performance defining method being applicable to the smooth choma of multistage centrifugal pump.

Background technique

The dynamic performance of the smooth choma of existing high pressure multiple-stage centrifugal pump is determined mostly rely on traditional short encapsulating method or utilize cfdrc to carry out.Short sealing method for solving have ignored the tangential velocity near bi-side in solution procedure, has carried out simplification degree comparatively greatly, have larger narrow limitation when solving slenderness ratio and being greater than the control ring seal of 0.25 to sealing fluid movement equation; Cfdrc analytical method is higher for the requirement of engineering calculation personnel, multiple step such as need modeling, solve, and choosing of turbulence model has stronger professional, and computing cycle is long.

Summary of the invention

The object of the invention is the dynamic performance defining method providing the smooth choma of a kind of high pressure multiple-stage centrifugal pump, described method comprises the steps: step one, utilize dimensional measuring instrument to determine the structural parameter of required high pressure multiple-stage centrifugal pump impeller port ring, and determine the operation operating mode of centrifugal pump; Step 2, sets up the governing equation group that equation of continuity, circumferential momentum equation and axial momentum equation form; Step 3, carries out dimensionless simplification to above-mentioned governing equation group and utilizes perturbation method to ask for full scale equation group perturbing about the single order of Perturbation and zeroth order the difference set of equation of form; Step 4, utilizes shooting method and solves dimensionless difference set of equation after simplification in conjunction with actual convergence criterion, and finally determining the size of high pressure multiple-stage centrifugal pump choma dynamic coefficients

Accurately can calculate slenderness ratio according to the dynamic performance defining method of the smooth choma of high pressure multiple-stage centrifugal pump of the present invention and be less than dynamics coefficient under the different operating mode of high pressure multiple-stage centrifugal pump impeller port ring (different operating mode refers to different rotating speeds for same pump and different lifting pump) of 0.5.And more adequately can ask for the dynamic response factor of smooth seal rotor-support-foundation system, the method Applicable scope is comparatively wide, and required computational resource is less, and engineer applied is worth larger.

Simultaneously, dynamic performance defining method according to the smooth choma of high pressure multiple-stage centrifugal pump of the present invention can apply to assess the high pressure multiple-stage centrifugal pump operation stability of conveying liquid body medium, the axle system of this high pressure multiple-stage centrifugal pump under declared working condition and the relation of single order wet criticality rotating speed can be judged by calculating relevant dynamic response factor, and then judge axle system whether in controlled surplus stability and safety run, security of operation and the stability of high pressure multiple-stage centrifugal pump can be put forward.

And, dynamic performance defining method according to the smooth choma of high pressure multiple-stage centrifugal pump of the present invention is applicable to the different geometrical size of smooth rotor sealing system and all operating modes, can calculate corresponding dynamic response factor according to different operation operating modes or structural parameter by method for solving.

Accompanying drawing explanation

Fig. 1 is the FB(flow block) of the dynamic performance defining method according to the smooth choma of high pressure multiple-stage centrifugal pump of the present invention of the present invention.

Embodiment

In order to make object of the present invention, technological scheme and advantage clearly understand, below the present invention is further elaborated.Should be appreciated that specific embodiment described herein only in order to explain the present invention instead of limitation of the present invention.

Accompanying drawing 1 is the FB(flow block) of the dynamic performance defining method according to the smooth choma of high pressure multiple-stage centrifugal pump of the present invention of the present invention.Below in conjunction with accompanying drawing 1, the present invention will be further described.

The present invention is applicable to the calculating of the dynamics coefficient of smooth rotor sealing system under all operating modes of different geometrical size, and this example is chosen certain smooth seal rotor and calculated under its declared working condition, the structural parameter of rotor sealing system:

C _r=0.1905mm,R=0.0762m,L=0.0762mm

Wherein, Cr represents seal clearance, and R represents the rotor internal diameter of sealing place, and L represents seal action length.

The operation operating mode of rotor sealing system: ω=3600r/min; Δ p=3.44MPa;

Media property: μ=1.295 × 10 ^-3ns/m ²; ρ=1000kg/m ³;

Step 2, set up the governing equation group of control volume model:

Control volume circumference momentum equation:

$\begin{array}{c}-\frac{h^2}{\mathrm{\μU}}\frac{\∂p}{\∂x}{\left(\frac{\mathrm{\μ}}{\mathrm{\ρUh}}\right)}^{1+m_0}=1/2n_0\{U_x{({U_x}^2+{U_y}^2)}^{\frac{1+m_0}{2}}+(U_x-1){({(U_x-1)}^2+{U_y}^2)}^{\frac{1+m_0}{2}}\}\\ +{\left(\frac{\mathrm{\μ}}{\mathrm{\ρU}{h}_0}\right)}^{m_0}\{\frac{h}{U}\frac{\∂U_x}{\∂t}+h{U}_x\frac{\∂U_x}{\∂x}+h{U}_y\frac{\∂U_x}{\∂y}\}\end{array}$

Control volume axial momentum equation:

$\begin{array}{c}-\frac{h^2}{\mathrm{\μU}}\frac{\∂p}{\∂y}{\left(\frac{\mathrm{\μ}}{\mathrm{\ρUh}}\right)}^{1+m_0}=1/2n_0\{U_y{({U_x}^2+{U_y}^2)}^{\frac{1+m_0}{2}}+U_y{({(U_x-1)}^2+{U_y}^2)}^{\frac{1+m_0}{2}}\}\\ +{\left(\frac{\mathrm{\μ}}{\mathrm{\ρU}{h}_0}\right)}^{m_0}\{\frac{h}{U}\frac{\∂U_y}{\∂t}+h{U}_x\frac{\∂U_y}{\∂x}+h{U}_y\frac{\∂U_y}{\∂y}\}\end{array}$

Control volume equation of continuity:

$H\frac{\∂u_z}{\∂z}+\frac{1}{R}\frac{\∂}{\∂\mathrm{\θ}}\left(H{u}_{\mathrm{\θ}}\right)+\frac{1}{\mathrm{R\ω}}\frac{\∂H}{\∂t}=0$

Step 3, governing equation group is carried out dimensionless simplification and obtain following Non-di-mensional equation group:

$\begin{array}{c}-\frac{H^2}{\mathrm{\μU}}\frac{\∂p}{\∂z}=1/2n_0{R_C}^{1+m_0}\{u_z{({u_{\mathrm{\θ}}}^2+{u_z}^2)}^{\frac{1+m_0}{2}}+u_z{({(u_{\mathrm{\θ}}-1)}^2+{u_z}^2)}^{\frac{1+m_0}{2}}\}\\ +R_C\{\frac{H}{U}\frac{\∂u_z}{\∂t}+H\frac{u_{\mathrm{\θ}}}{R}\frac{\∂u_z}{\∂x}+H{u}_z\frac{\∂u_z}{\∂z}\}\end{array}$

$\begin{array}{c}-\frac{H^2}{\mathrm{\μU}}\frac{1}{R}\frac{\∂p}{\∂\mathrm{\θ}}=1/2n_0{R_C}^{1+m_0}\{u_{\mathrm{\θ}}{({u_{\mathrm{\θ}}}^2+{u_z}^2)}^{\frac{1+m_0}{2}}+(u_{\mathrm{\θ}}-1){({(u_{\mathrm{\θ}}-1)}^2+{u_z}^2)}^{\frac{1+m_0}{2}}\}\\ +R_C\{\frac{H}{U}\frac{\∂u_{\mathrm{\θ}}}{\∂t}+H\frac{u_{\mathrm{\θ}}}{R}\frac{\∂u_{\mathrm{\θ}}}{\∂\mathrm{\θ}}+H{u}_z\frac{\∂u_{\mathrm{\θ}}}{\∂y}\}\end{array}$

$H\frac{\∂u_z}{\∂z}+\frac{1}{R}\frac{\∂}{\∂\mathrm{\θ}}\left(H{u}_{\mathrm{\θ}}\right)+\frac{1}{\mathrm{R\ω}}\frac{\∂H}{\∂t}=0$

Step 4, introducing disturbance displacement under axle vortex motion state is Perturbation ε, and represents the zeroth order about this Perturbation and the single order form of the circumference of choma inner fluid, axial velocity, pressure and gap width respectively:

u _z=u _z0+εu _z1,H=C+εH ₁

u _θ=u _θ0+εu _θ1,p=p ₀+εp ₁

Ask for Non-di-mensional equation group by perturbation method to perturb about the single order of Perturbation and zeroth order the difference set of equation of form:

$\frac{\∂u_{\mathrm{\θ}1}}{\∂z}+\[\mathrm{\σ}{B}_3-\mathrm{j\ωT}(\frac{1}{2}+v)\]u_{\mathrm{\θ}1}+\frac{\∂u_{\mathrm{\θ}1}}{\∂\mathrm{\τ}}-\mathrm{b\σ}{B}_{4}v{u}_{z1}-\mathrm{jb}\left(\frac{L}{R}\right)p_1=-\mathrm{\σ}{B}_{5}v\left(\frac{h_1}{\mathrm{\ϵ}}\right)$

$\frac{\∂u_z}{\∂z}+b\frac{\∂p_1}{\∂z}+\[\mathrm{\σ}{B}_1-\mathrm{j\ωT}(\frac{1}{2}+v)\]u_{z1}+\frac{\∂u_z}{\∂\mathrm{\τ}}+\mathrm{b\σ}{B}_{2}v{u}_{\mathrm{\θ}1}=-(1-m_0)\mathrm{b\σ}\left(\frac{h_1}{\mathrm{\ϵ}}\right)$

$\frac{\∂u_{z1}}{\∂z}-j\left(\frac{L}{R}\right)u_{\mathrm{\θ}1}=-j\left(\frac{L}{R}\right)(\frac{1}{2}+v)\left(\frac{h_1}{\mathrm{\ϵ}}\right)+b\frac{\∂}{\∂\mathrm{\τ}}\left(\frac{h_1}{\mathrm{\ϵ}}\right)$

Above difference set of equation simplified, after simplifying, difference set of equation is:

$\frac{d}{\mathrm{dz}}\left\{\begin{array}{c}{u}_{z1}\\ u_{\mathrm{\θ}1}\\ p_1\end{array}\right\}+\left[\begin{array}{ccc}0& -j\frac{L}{R}& 0\\ -\mathrm{b\σ}{B}_{4}v& \mathrm{\σ}{B}_3+\mathrm{j\ΓT}& -\mathrm{jb}\frac{L}{R}\\ (\mathrm{\σ}{B}_1+\mathrm{j\ΓT})/b& \mathrm{\σ}{B}_{2}v+\mathrm{j\ωT}& 0\end{array}\right]\left\{\begin{array}{c}{u}_{z1}\\ u_{\mathrm{\θ}1}\\ p_1\end{array}\right\}=\frac{r_0}{\mathrm{\ϵ}}\left\{\begin{array}{c}\mathrm{jb}\[\mathrm{\ΩT}-\mathrm{\ωT}(\frac{1}{2}+v)\]\\ -\mathrm{\σ}{B}_{5}v\\ -(1-m_0)\mathrm{\σ}-j\[\mathrm{\ΩT}-\mathrm{\ωT}(\frac{1}{2}+v)\])\end{array}\right\}$

And the initial value Solve problems this problem incorporated into into partial differential equation, and utilize Newton method to improve initial value, improve one's methods as follows:

$u_{z10}=u_{z10}-\frac{F\left(u_{z10}\right)}{F^{\′}\left(u_{z10}\right)}$

Wherein, p is met ₁(L)=0; u _{θ 1}(0)=0; p ₁(0)=-(1+ ξ) u _z10the F function of/b is:

$F\left(u_{z10}\right)=\frac{d}{\mathrm{dz}}\left\{\begin{array}{c}{u}_{z1}\\ u_{\mathrm{\θ}1}\\ p_1\end{array}\right\}+\left[\begin{array}{ccc}0& -j\frac{L}{R}& 0\\ -\mathrm{b\σ}{B}_{4}v& \mathrm{\σ}{B}_3+\mathrm{j\ΓT}& -\mathrm{jb}\frac{L}{R}\\ (\mathrm{\σ}{B}_1+\mathrm{j\ΓT})/b& \mathrm{\σ}{B}_{2}v+\mathrm{j\ωT}& 0\end{array}\right]\left\{\begin{array}{c}{u}_{z1}\\ u_{\mathrm{\θ}1}\\ p_1\end{array}\right\}-\frac{r_0}{\mathrm{\ϵ}}\left\{\begin{array}{c}\mathrm{jb}\[\mathrm{\ΩT}-\mathrm{\ωT}(\frac{1}{2}+v)\]\\ -\mathrm{\σ}{B}_{5}v\\ -(1-m_0)\mathrm{\σ}-j\[\mathrm{\ΩT}-\mathrm{\ωT}(\frac{1}{2}+v)\])\end{array}\right\}$

F ' function is:

$F^{\′}\left(u_{z10}\right)=\frac{d}{\mathrm{dz}}\left\{\begin{array}{c}{u}_{z1}\\ u_{\mathrm{\θ}1}\\ p_1\end{array}\right\}+\left[\begin{array}{ccc}0& -j\frac{L}{R}& 0\\ -\mathrm{b\σ}{B}_{4}v& \mathrm{\σ}{B}_3+\mathrm{j\ΓT}& -\mathrm{jb}\frac{L}{R}\\ (\mathrm{\σ}{B}_1+\mathrm{j\ΓT})/b& \mathrm{\σ}{B}_{2}v+\mathrm{j\ωT}& 0\end{array}\right]\left\{\begin{array}{c}{u}_{z1}\\ u_{\mathrm{\θ}1}\\ p_1\end{array}\right\}$

Utilize shooting method to simplify difference set of equation to above dimensionless and carry out numerical solution, until complete p ₁(L)=0; u _{θ 1}(0)=0; p ₁(0)=-(1+ ξ) u _z10the condition of convergence of/b.Gained numerical result is respectively three groups of axial, circumference with the pressure size plural arrays corresponding from different z coordinate, carries out matching respectively, obtain real part functions F to the real part of pressure array and imaginary part ₃(Ω) with imaginary part function F ₄(Ω), after this example fits, function analytic expression is:

f ₃(Ω ₁)=-0.0093z ⁴+0.1182z ³+0.4853z ²-1.1214z+0.6639

f ₄(Ω ₁)=-0.1252z ⁴+0.2345z ³-1.3113z ²+1.0052z+0.2309

Separately get foundation and solution procedure that two different eddy velocity Ω repeat above governing equation group, the analytic expression of the real part functions asked and imaginary part function is:

f ₃(Ω ₂)=0.0207z ⁴-0.1732z ³+0.8532z ²-1.4595z+0.6593

f ₄(Ω ₂)=-0.2510z ⁴+0.4694z ³-2.6223z ²+2.0096z+0.4625

f ₃(Ω ₃)=0.0703z ⁴-0.2647z ³+1.4665z ²-2.0235z+0.6517

f ₄(Ω ₃)=-0.3777z ⁴+0.7050z ³-3.9329z ²+3.0124z+0.6953

The calculation expression of high pressure multiple-stage centrifugal pump choma dynamic coefficients is:

$\begin{array}{c}K=F_2\left({\mathrm{\Ω}}_1\right)\·\frac{(F_3\left({\mathrm{\Ω}}_1\right)-F_3\left({\mathrm{\Ω}}_2\right))(F_1\left({\mathrm{\Ω}}_1\right)-F_1\left({\mathrm{\Ω}}_3\right))-(F_3\left({\mathrm{\Ω}}_1\right)-F_3\left({\mathrm{\Ω}}_3\right))(F_1\left({\mathrm{\Ω}}_1\right)-F_1\left({\mathrm{\Ω}}_2\right))}{(F_2\left({\mathrm{\Ω}}_1\right)-F_2\left({\mathrm{\Ω}}_3\right))(F_1\left({\mathrm{\Ω}}_1\right)-F_1\left({\mathrm{\Ω}}_2\right))-(F_2\left({\mathrm{\Ω}}_1\right)-F_2\left({\mathrm{\Ω}}_2\right))(F_1\left({\mathrm{\Ω}}_1\right)-F_1\left({\mathrm{\Ω}}_3\right))}\\ +F_3\left({\mathrm{\Ω}}_1\right)-F_1\left({\mathrm{\Ω}}_1\right)\·\frac{(F_3\left({\mathrm{\Ω}}_1\right)-F_3\left({\mathrm{\Ω}}_2\right))(F_2\left({\mathrm{\Ω}}_1\right)-F_2\left({\mathrm{\Ω}}_3\right))-(F_3\left({\mathrm{\Ω}}_1\right)-F_3\left({\mathrm{\Ω}}_3\right))(F_2\left({\mathrm{\Ω}}_1\right)-F_2\left({\mathrm{\Ω}}_2\right))}{(F_2\left({\mathrm{\Ω}}_1\right)-F_2\left({\mathrm{\Ω}}_3\right))(F_1\left({\mathrm{\Ω}}_1\right)-F_1\left({\mathrm{\Ω}}_2\right))-(F_2\left({\mathrm{\Ω}}_1\right)-F_2\left({\mathrm{\Ω}}_2\right))(F_1\left({\mathrm{\Ω}}_1\right)-F_1\left({\mathrm{\Ω}}_3\right))}\end{array}$

$k=F_4\left({\mathrm{\Ω}}_1\right)-\frac{F_4\left({\mathrm{\Ω}}_1\right)-F_4\left({\mathrm{\Ω}}_2\right)}{F_2\left({\mathrm{\Ω}}_1\right)-F_2\left({\mathrm{\Ω}}_2\right)}{F}_2\left({\mathrm{\Ω}}_1\right)$

$C=-\frac{F_4\left({\mathrm{\Ω}}_1\right)-F_4\left({\mathrm{\Ω}}_2\right)}{F_2\left({\mathrm{\Ω}}_1\right)-F_2\left({\mathrm{\Ω}}_2\right)}$

$c=\frac{(F_3\left({\mathrm{\Ω}}_1\right)-F_3\left({\mathrm{\Ω}}_2\right))(F_2\left({\mathrm{\Ω}}_1\right)-F_2\left({\mathrm{\Ω}}_3\right))-(F_3\left({\mathrm{\Ω}}_1\right)-F_3\left({\mathrm{\Ω}}_3\right))(F_2\left({\mathrm{\Ω}}_1\right)-F_2\left({\mathrm{\Ω}}_2\right))}{(F_2\left({\mathrm{\Ω}}_1\right)-F_2\left({\mathrm{\Ω}}_3\right))(F_1\left({\mathrm{\Ω}}_1\right)-F_1\left({\mathrm{\Ω}}_2\right))-(F_2\left({\mathrm{\Ω}}_1\right)-F_2\left({\mathrm{\Ω}}_2\right))(F_1\left({\mathrm{\Ω}}_1\right)-F_1\left({\mathrm{\Ω}}_3\right))}$

$M=\frac{(F_3\left({\mathrm{\Ω}}_1\right)-F_3\left({\mathrm{\Ω}}_2\right))(F_1\left({\mathrm{\Ω}}_1\right)-F_1\left({\mathrm{\Ω}}_3\right))-(F_3\left({\mathrm{\Ω}}_1\right)-F_3\left({\mathrm{\Ω}}_3\right))(F_1\left({\mathrm{\Ω}}_1\right)-F_1\left({\mathrm{\Ω}}_2\right))}{(F_2\left({\mathrm{\Ω}}_1\right)-F_2\left({\mathrm{\Ω}}_3\right))(F_1\left({\mathrm{\Ω}}_1\right)-F_1\left({\mathrm{\Ω}}_2\right))-(F_2\left({\mathrm{\Ω}}_1\right)-F_2\left({\mathrm{\Ω}}_2\right))(F_1\left({\mathrm{\Ω}}_1\right)-F_1\left({\mathrm{\Ω}}_3\right))}$

Wherein, K, k, C, c, M are respectively Main rigidity, intersection rigidity, main damping, intersection damping and main associated mass, and correlation function is as follows:

$F_1\left(\mathrm{\Ω}\right)=\left(\mathrm{\ΩT}\right);F_2\left(\mathrm{\Ω}\right)={\left(\mathrm{\ΩT}\right)}^2;F_3\left(\mathrm{\Ω}\right)=\underset{0}{\overset{1}{\∫}}{f}_3\left(\mathrm{\Ω}\right)\mathrm{d\Ω};F_4\left(\mathrm{\Ω}\right)=\underset{0}{\overset{1}{\∫}}{f}_4\left(\mathrm{\Ω}\right)\mathrm{d\Ω};$

Although describe the present invention with reference to preferred embodiment, those skilled in the art will recognize, can carry out the change in form and details, only otherwise disengaging the spirit and scope of the present invention.The present invention attempts to be not limited to the specific embodiment be disclosed, and as expected for implementing optimal mode of the present invention, on the contrary, the present invention will comprise whole embodiments of the scope falling into accessory claim.

Claims (6)

1. a dynamic performance defining method for the smooth choma of high pressure multiple-stage centrifugal pump, described method comprises the steps:

Step one, utilizes dimensional measuring instrument to determine the structural parameter of required high pressure multiple-stage centrifugal pump impeller port ring, and determines the operation operating mode of centrifugal pump;

Step 2, sets up the governing equation group that equation of continuity, circumferential momentum equation and axial momentum equation form;

Step 3, carries out dimensionless simplification to above-mentioned governing equation group and utilizes perturbation method to ask for full scale equation group perturbing about the single order of Perturbation and zeroth order the difference set of equation of form;

Step 4, utilizes shooting method and solves dimensionless difference set of equation after simplification in conjunction with actual convergence criterion, and finally determining the size of high pressure multiple-stage centrifugal pump choma dynamic coefficients;

In described step 4, shooting method is adopted to ask for the numerical solution of original error value governing equation group under its border condition of convergence and matching completes its analytic solutions representation, in solution procedure, newton's initial value is adopted to improve one's methods, first to original error value set of equation about the overall differentiate of z, ask for the numerical solution of derivative equation group, derivative equation group reduced form is as follows:

2. the dynamic performance defining method of the smooth choma of high pressure multiple-stage centrifugal pump according to claim 1, is characterized in that described method is applicable to the dynamic response of high pressure multiple-stage centrifugal pump impeller port ring under arbitrary operating mode that slenderness ratio is less than or equal to 0.5 and determines.

3. the dynamic performance defining method of the smooth choma of high pressure multiple-stage centrifugal pump according to claim 1, it is characterized in that, in described step 2, control volume set of equation is made up of axial, the circumferential momentum equation of control volume and equation of continuity, and concrete form is:

Circumference momentum equation:

Axial momentum equation:

Equation of continuity:

4. the dynamic performance defining method of the smooth choma of high pressure multiple-stage centrifugal pump according to claim 1, is characterized in that, in described step 3, after dimensionless process, governing equation group is:

5. the dynamic performance defining method of the smooth choma of high pressure multiple-stage centrifugal pump according to claim 1, it is characterized in that, in described step 3, under introducing axle vortex motion state, disturbance displacement is Perturbation, and represent the zeroth order about this Perturbation and the single order form of the circumference of choma inner fluid, axial velocity, pressure and gap width respectively, and then obtain the single order of former Non-di-mensional equation group about this Perturbation and the difference set of equation of zeroth order form:

6. the dynamic performance defining method of the smooth choma of high pressure multiple-stage centrifugal pump according to claim 1, is characterized in that, in described step 4, the calculation expression of gained high pressure multiple-stage centrifugal pump choma dynamic coefficients is:

Wherein, K, k, C, c, M are respectively Main rigidity, intersection rigidity, main damping, intersection damping and main associated mass, function F ₁, F ₂, F ₃, F ₄be respectively the function relevant to eddy velocity.