An estimation of the velocity profile for the laminar-turbulent transition in the plane jet on the basis of the theory of stochastic equations and equivalence of measures

Abstract

The theory of stochastic equations and the theory of equivalence of measures previously applied to flows in the boundary layer and in the pipe are considered to calculate the velocity profile of the flat jet. This theory previously made it possible to determine the critical Reynolds number and the critical point for the flow of the plane jet. Here based on these results the analytical dependence for the index of the velocity profile is derived. Velocity profiles are calculated for a laminar-turbulent transition in the jet. This formula reliably reflects an increase of the energy transferred from a deterministic state to a random one with an increase of the index of the velocity profile. Results show satisfactory agreement with the known experimental data for the velocity profile of the flat jet. Using obtained results it is possible to determine the location of technical devices for laminarization of the flow in the jet. This is important both for reducing friction in the flow around aerodynamic vehicles and for maintaining the jet profile if it is necessary to ensure the stability of the flow characteristics. Also the obtained relations can be useful for researching of the processes in combustion chambers, in the case of welding and in other technical devices.

1 Introduction

The ideas of the theory of the onset of turbulence [1,2,3,4,5,6,7,8,9,10,11,12,13] both theoretical and numerical [14,15,16,17,18,19,20,21,22,23], including also the statistic and DNS modeling [2, 24,25,26,27,28,29,30,31,32], provide the necessity for determination of the stochastic equations and the regularity of equivalence of measures. Investigations of the onset of turbulence based on the theories of stochastic equations and the equivalence of measures between deterministic and random motion were carried out mainly for flows in shear flows in a circular tube [33,34,35,36,37,38,39,40,41,42,43], on a flat smooth plate [44,45,46,47,48,49,50,51,52,53,54,55], as well as for motion near a rotating disk [56] and for motion between rotating coaxial cylinders [57]. The last research was done for laminar–turbulent transition on a flat plate [58, 59]. All of these publications refer to shear flows in the presence of a solid surface (wall). An article related to forced shear flow in the absence of a wall determined the critical Reynolds number and the critical point for a plane jet flow [60]. In this regard, it seems important to present a procedure for deriving the velocity profile parameters for a plane jet flow during the transition from laminar to turbulent motion. The problem of the transition of laminar flow in a jet to turbulent is discussed in articles [60,61,62,63,64,65,66,67,68]. Note that in accordance to the opinions of works [61, 64], the problem of the transition to turbulence in the jet does not cause such a keen interest because the transition has already happened when critical Reynolds numbers equal to 30. In this case, the Reynolds number is made up of the entire width of a flat slit. In the case where the Reynolds number is made up using half the width of a flat slit, it equals 5–10. In this article, the procedure of calculation of parameters velocity profile of flat jet is considered on the basis of the theory of stochastic equations and the theory of equivalence of measures. The analytical dependence for the index of the velocity profile has the right and the left sides. The right side of the equation includes turbulent Reynolds numbers. The left side of the equation, which includes the desired profile index, is determined by the parameters of the averaged motion values and Reynolds number. As a result, having calculated the left side of the formula from the averaged values of the motion at a fixed velocity profile index, we obtain the result and compare it with the right side of the equation, which is determined by the experimental turbulent Reynolds number. If both parts of the equation agree satisfactorily, we compare the given speed profile index with its values from the known experimental range.

2 Conservation stochastic equations

Conservation stochastic equations were derived in [33,34,35,36,37,38,39,40,41,42,43]:

the equation of continuity

$$\frac{d{\left(\rho \right)}_{col_{st}}}{d\tau}=-\frac{{\left(\rho \right)}_{st}}{\tau_{cor}}-\frac{d{\left(\rho \right)}_{st}}{d\tau},$$

(1)

the momentum equation

$$\frac{d{\left(\rho {u}_i\right)}_{col_{st}}}{d\tau}={div}{\left({\tau}_{i,j}\right)}_{col_{st}}+{div}{\left({\tau}_{i,j}\right)}_{st}-\frac{{\left(\rho {u}_i\right)}_{st}}{\tau_{cor}}-\frac{d{\left(\rho {u}_i\right)}_{st}}{d\tau}+{F}_{col_{st}}+{F}_{st},$$

(2)

and the energy equation

$$\frac{{d E}_{col_{st}}}{d\tau}={div}{\left(\lambda \frac{\partial T}{\partial {x}_j}+{u}_i{\tau}_{i,j}\right)}_{col_{st}}+{div}{\left(\lambda \frac{\partial T}{\partial {x}_j}+{u}_i{\tau}_{i,j}\right)}_{st}-\left(\frac{E_{st}}{\tau_{cor}}\right)-\left(\frac{{d E}_{st}}{d\tau}\right)+{\left({u}_iF\right)}_{col_{st}}+{\left({u}_iF\right)}_{st}.$$

(3)

Here, $E,\rho, \overrightarrow{U},{u}_i,{u}_j,{u}_l,\mu, \tau, {\tau}_{i,j}$ are the energy, the density; the velocity vector; the velocity components in directions x_i, x_j, x_l (i, j, l = 1, 2, 3); the dynamic viscosity; the time; and the stress tensor τ_i,j = P + σ_i,j, ${\sigma}_{ij}=\mu \left(\frac{\partial {u}_i}{\partial {x}_j}+\frac{\partial {u}_j}{\partial {x}_i}\right)-{\delta}_{ij}\left(\xi -\frac{2}{3}\mu \right)\frac{\partial {u}_l}{\partial {x}_l}$, ${{\tau}_{cor}=\frac{L}{{\left({\left({E}_{st}\right)}_{U,P}/\rho \right)}^{1/2}}}$, δ_ij = 1 if i = j, δ_ij = 0 for i ≠ j. Р is the pressure of liquid or gas; λ is the thermal conductivity; c_p and c_v are the specific heat at constant pressure and volume, respectively; F is the external force. Further, L = L_{U, P} = L_U is the scale of turbulence. Indexes (_{U, P}) and (_U) refer to the velocity field. L_y on x₂ = y, or L_x, x₁ = x. Here, x₁ and x₂ are coordinates along and normal to the axis of the jet. Index “col_st” refers to components, which are actually the deterministic. Index “st” refers to component, which are actually the stochastic. Then for the non-isothermal motion of the medium, using the definition of equivalency measures between deterministic and random process in the critical point, the sets of stochastic equations of energy, momentum, and mass are defined for the next space-time areas: 1) the beginning of the generation (index 1,0 or 1); 2) generation (index 1,1); 3) diffusion (1,1,1) and 4) the dissipation of the turbulent fields. These results provide an opportunity to introduce the concept of the correlator, which is defined for the potential physical quantities and combinations (N, M). This correlator, in its structure, will determine the possible range of motion in space depending on the different combinations of (M, N) and the corresponding values ​​that determine the correlation interval of space-time. According to [2, 32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50], the correlator in space-time is

$${m}_i\to {m}_c;{r}_i\overset{\lim }{\to }{r}_c;\Delta {\tau}_i\to {\tau}_c\left({D}_{N,M}\left({m}_i;{r}_i;\Delta {\tau}_i\right)\right)=0,$$

(4)

$$D_{N,M}\left(m_c;r_c;\tau_c\right)={\underset i\Sigma}_{m_i\xrightarrow\lim m_cr_i\xrightarrow\lim rc\Delta\tau_i\xrightarrow\lim\tau_c}\left\{m\left(T^MZ^\ast\cap T^NY^\ast\right)-R_{{\text{l}}_{T^MZ^\ast T^NY^\ast}}m\left(T^M\text{Z}^\ast\right)\right\}.$$

(5)

Subscript j denotes the parameters m_cj (j = 3 means mass, momentum, and energy). For the case of the binary intersections, it was written that X = Y + Z + W. Here subscripts cr or c refer to critical point r(x_cr, τ_cr) or r_c: the space-time point of the beginning of the interaction between the deterministic field and random field that leads to turbulence. In addition, subsets Y, Z, W are сalled extended in X. For the transfer of the substantial quantity Φ (mass (density ρ), momentum (ρU), energy (E)) of the deterministic (laminar) motion into a random (turbulent) one, for domain 1 of the start of turbulence generation, pair (N, M) = (1, 0), with the equivalence of measures being written as (${d\Phi}_{col_{st}}$)_1,0 =  − R_1,0(Φ_st) and ${\left(\frac{d{\left(\Phi \right)}_{{col}_{st}}}{d\tau}\right)}_{1,0}=-{R}_{1,0}\left(\frac{\Phi_{st}}{\tau_{cor}}\right)$. Applying correlation D_N,M(r_c; m_ci; τ_c) = D_1,1(r_c; m_ci, τ_c) derived in [2, 32,33,34,35,36,37,38,39,40,41,42], the equivalence relation for pair (N, M) = (1, 1) was defined as (${d\Phi}_{col_{st}}$)_1,1 =  − R_1,1(dΦ_st), ${\left(\frac{d{\left(\Phi \right)}_{col_{st}}}{d\tau}\right)}_{1,1}=-{R}_{1,1}\left(\frac{d{\Phi}_{st}}{d\tau}\right)$, where R_1,0 and R_1,1 are fractal coefficients, $\Phi_{col_{st}}$ is the part of the field of Φ, notably, its deterministic component (subscript col_st) is the stochastic component of the measure, which is zero; Φ_st is the part of Φ, notably, the proper stochastic component (subscript st).

3 Critical Reynolds number and critical point

In article [58] it was shown that the critical Reynolds number and critical point for the plane jet were determined using the set of stochastic Eqs. (1)–(3) for the area 1), which, referring the pair (N, M) = (1, 0) is:

$${\left(\frac{d\left(\rho \right){col}_{st}}{d\tau}\right)}_{1,0}=-\frac{\rho_{st}}{\tau_{cor}};$$

$$\left\{\begin{array}{l}{\left(\frac{d{\left(\rho \overrightarrow{U}\right)}_{col_{st}}}{d\tau}\right)}_{1,0}=-\left(\frac{{\left(\rho \overrightarrow{U}\right)}_{st}}{\tau_{cor}}\right);\\ {}\mathit{div}{\left({\tau}_{i,j}\right)}_{col_{st0}}=\frac{{\left(\rho \overrightarrow{U}\right)}_{st}}{\tau_{cor}}.\end{array}\right.$$

(6)

$$\left\{\begin{array}{l}{\left(\frac{d{(E)}_{{col}_{st}}}{d\tau}\right)}_{1,0}=-{\left(\frac{(E)_{st}}{\tau_{cor}}\right)}_{1,0};\\ {}{div}{\left(\lambda \frac{\partial T}{\partial {x}_j}+{u}_i{\tau}_{i,j}\right)}_{col_{st0}}={\left(\frac{(E)_{st}}{\tau_{cor}}\right)}_{1,0}.\end{array}\right.$$

So, the formula for critical Reynolds number was written as

$${\left({Re}\right)}_{{cr}}=21.1{\left(\frac{{{U}}_{{m}}}{\sqrt{{{E}}_{{st}}/\rho }}\right)}^{0.568}{\left(\frac{{L}}{x_2}\right)}^{0.973}{\left(\frac{{h}}{{L}}\right)}^{0.243}$$

(7)

and in according with data [52,53,54,55,56,57], the theoretical value for critical Reynolds number is

$${\left({Re}\right)}_{{cr}}=7.6\div 25.$$

(8)

U_m is the average velocity on the jet axis, h the width of the initial section of the jet. The definition of the value of the critical point is found from the equation $\int\limits_{-\Delta V\left|2\right.}^{+\Delta V\left|2\right.}d{\left({{E}}_{col_{st}}\right)}_{1,0}=\underset{X}{\int }{dE}_{st}$. Here E_st is the stochastic energy component in the space X with the measure m(E_st) < ∞ and ${E}_{st}={E}_{st}\left(\overrightarrow{x_i},{\tau}_i,{m}_i\right)<\infty$. According to the main results of the ergodic theory we have $\underset{X}{\int }{dE}_{st}=\frac{1}{\Delta V}\underset{V}{\int }{E}_{st}\delta \left({\left(\Delta V\right)}_{critic}-\Delta V\right) d V=\frac{1}{\tau_{cor}^0}\underset{\tau }\int {E}_{st}\delta \left({\tau}_{cor}^0-\tau \right) d\tau ={\left({E}_{st}\right)}_{critic}$_. (E_st)_critic is the stochastic energy at the critical point. Then it is possible to write as $\underset{X}{\int }{dE}_{st}=\frac{1}{L}\underset{L}{\int }{E}_{st}\delta \left({\left({{x}}_i\right)}_{critic}-{{x}}_i\right) d L=\frac{1}{\tau_{cor}^0}\underset{\tau }{\int }{E}_{st}\delta \left({\tau}_{cor}^0-\tau \right) d\tau ={\left({E}_{st}\right)}_{critic}$, here L is the scale of disturbance. So using these formulas, the equation for the critical point in the plane jet was determined [58]:

$${\left(\frac{{x}_2}{{x}_1}\right)}_{cr}={\left(1.75\right)}^{-10/9}\cdot {\left(\frac{{U}_{m}}{\sqrt{{E}_{st}/\rho }}\right)}^{-4/3}{\left(\frac{L}{x_2}\right)}^{-10/9}{\left(\frac{h}{L}\right)}^{-10/37},$$

(9)

and in according with data [52,53,54,55,56,57,58], the theoretical value for critical point ${\left(\frac{{x}_2}{{x}_1}\right)}_{cr}$ is

$${\left(\frac{{x}_2}{{x}_1}\right)}_{cr}=0.025.$$

(10)

4 The velocity profile characteristics of the flat jet

The set of stochastic Eqs. (1)–(3) for the area 2), referring the pair (N,M) = (1,1) is:

$${\left(\frac{d{\left(\rho \right)}_{col_{st}}}{d\tau}\right)}_{1,1}=-\frac{d{\left(\rho \right)}_{st}}{d\tau};$$

$$\left\{\begin{array}{l}{\left(\frac{d{\left(\rho \overrightarrow{U}\right)}_{col_{st}}}{d\tau}\right)}_{1,1}=-\left(\frac{d{\left(\rho \overrightarrow{U}\right)}_{st}}{d\tau}\right);\\ {}{div}{\left({\tau}_{i,j}\right)}_{col_{st1}}=\frac{d{\left(\rho \overrightarrow{U}\right)}_{st}}{d\tau}.\end{array}\right.$$

(11)

$$\left\{\begin{array}{l}{\left(\frac{d{(E)}_{{col}_{st}}}{d\tau}\right)}_{1,1}=-{\left(\frac{d{(E)}_{st}}{d\tau}\right)}_{1,1};\\ {}{div}{\left(\lambda \frac{\partial T}{\partial {x}_j}+{u}_i{\tau}_{i,j}\right)}_{col_{st1}}={\left(\frac{d{(E)}_{st}}{d\tau}\right)}_{1,1}.\end{array}\right.$$

Then, taking into account the definition of the velocity u₁ of laminar motion in a plane jet [61, 64].

$${u}_1=0.4543{\left(\frac{{\left({U}_m^2h\right)}^2}{\nu \cdot {x}_1}\right)}^{1/3}\left(1-{\left( th\left(\xi \right)\right)}^2\right).$$

(12)

Taking into account that

$$\xi =0.2752{\left(\frac{\left({U}_m^2h\right)}{\nu^2}\right)}^{1/3}\frac{x_2}{x_1^{2/3}}$$

(13)

and introducing the relation

$$\frac{th\left(\xi \right)}{ch\left(\xi \right)}=\frac{\xi -\frac{\xi^3}{3}+\frac{2{\xi}^5}{15}.\dots }{1+\frac{\xi^2}{2!}+\frac{\xi^4}{4!}.\dots }=f\left(\xi \right),$$

(14)

the derivative $\frac{du_1}{dx_1}$ is

$$\frac{{d u}_1}{{d x}_1}=-\frac{1}{3}0.4543{\left(\frac{{\left({U}_m^2h\right)}^2}{\nu \cdot {x}_1}\right)}^{1/3}\frac{1}{x_1}-0.4543{\left(\frac{{\left({U}_m^2h\right)}^2}{\nu \cdot {x}_1}\right)}^{1/3}\cdot 2f\left(\xi \right)\frac{d\xi}{{d x}_1},$$

(15)

$$\frac{d\xi}{{d x}_1}=\left[0.2752{\left(\frac{\left({U}_m^2h\right)}{\nu^2}\right)}^{1/3}\frac{x_2}{x_1^{2/3}}\right]\cdot \left(-\frac{2}{3}\frac{1}{x_1}\right)=-\frac{2}{3}\frac{1}{x_1}\cdot \xi,$$

(16)

$$\frac{du_1}{dx_1}=-\frac{1}{3}0.4543{\left(\frac{{\left({U}_m^2h\right)}^2}{\nu \cdot {x}_1}\right)}^{1/3}\frac{1}{x_1}-0.4543{\left(\frac{{\left({U}_m^2h\right)}^2}{\nu \cdot {x}_1}\right)}^{1/3}\cdot 2f\left(\xi \right)\left(-\frac{2}{3}\frac{1}{x_1}\cdot \xi \right)=0.4543{\left(\frac{{\left({U}_m^2h\right)}^2}{\nu \cdot {x}_1}\right)}^{1/3}\frac{1}{x_1}\left(-\frac{1}{3}+\frac{4}{3}\cdot {\xi}^2\right),$$

(17)

$${\left(\frac{du_1}{dx_1}\right)}^2={\left[0.4543{\left(\frac{{\left({U}_m^2h\right)}^2}{\nu \cdot {x}_1}\right)}^{1/3}\frac{1}{x_1}\left(-\frac{1}{3}+\frac{4}{3}\cdot {\xi}^2\right)\right]}^2.$$

(18)

In case of small values of ξ for laminar flow we have next formulas:

$${\left(\frac{du_1}{dx_1}\right)}^2={\left[0.4543{\left(\frac{{\left({U}_m^2h\right)}^2}{\nu \cdot {x}_1}\right)}^{1/3}\frac{1}{x_1}\left(-\frac{1}{3}\right)\right]}^2=\frac{1}{9}{\left[0.4543{\left(\frac{{\left({U}_m^2h\right)}^2}{\nu \cdot {x}_1}\right)}^{1/3}\frac{1}{x_1}\right]}^2,$$

(19)

$${\left(\frac{du_1}{dx_1}\right)}^2=\left[\frac{1}{9}{0.4543}^2{\left(\frac{{\left({U}_m^2h\right)}^2}{\nu \cdot {x}_1}\right)}^{1/3}{\left(\frac{{\left({U}_m^2h\right)}^2}{\nu \cdot {x}_1}\right)}^{1/3}\frac{1}{x_1^2}\right]=\left[\frac{1}{9}{0.4543}^2{\left(\frac{\left({U}_m^8{h}^4\right)}{x_1^2{\nu}^2}\right)}^{1/3}\frac{1}{x_1^2}\right],$$

(20)

$${\left(\frac{du_1}{dx_1}\right)}^2=\left[\frac{1}{9}{0.4543}^2{\left(\frac{\left({U}_m^8{h}^4\right)}{x_1^2{\nu}^2}\right)}^{1/3}\frac{1}{x_1^2}\right]=\left[\frac{1}{9}{0.4543}^2{U}_m^2{\left(\frac{h^2}{x_1^2}\right)}^{1/3}{\left(\frac{\left({U}_m^2{h}^2\right)}{\nu^2}\right)}^{1/3}\frac{1}{x_1^2}\right].$$

(21)

So finally we have

$$\mu {\left(\frac{du_1}{dx_1}\right)}^2=\rho \nu \left[\frac{1}{9}{0.4543}^2{U}_m^2{\left(\frac{h}{x_1}\right)}^{2/3}{\left({Re}\right)}^{2/3}\frac{1}{x_1^2}\right].$$

(22)

For the flow region of the beginning of turbulence initiation, we assume the same dependence, but with a different degree “n”. As is known, if the velocity u₁ in laminar motion is ~ (x₁)^-1/3, then for a developed turbulent region u₁ ~ (x₁)^-1/n and “n = 2”. Thus, for the transition region the values of degree “n” are in the interval 2 ≤ n ≤ 3. Next, we write down the relation determined earlier from the equivalence of measures for the region (1, 1) - the generation of turbulence [33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58]:

$$\frac{{\left[\rho \nu {\left(\frac{\partial {u}_1}{\partial {x}_2}\right)}^2\Delta \tau \right]}_{st och}}{{\left[\rho \nu {\left(\frac{\partial {u}_1}{\partial {x}_2}\right)}^2\Delta \tau \right]}_{lamin}}=\left|{{Re}_{st}}-\frac{1}{{Re}_{st}}\right|.$$

(23)

As it was shown in the papers [33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58], the Eq. (23) determines the relative increase of the energy transferred from the deterministic state to the random one. Subscript “lamin” refers to the laminar flow, and subscript “stoch” refers to the non-laminar flow. Taking into account that for transition regime formula (22) may be written as $\mu {\left(\frac{du_1}{dx_1}\right)}^2=\rho \nu \left[\frac{1}{n^2}{0.4543}^2{\left(\frac{U_m}{x_1}\right)}^2{\left(\frac{h}{x_1}\right)}^{2/n}{\left({Re}\right)}^{2/n}\right]$. Then in accordance with [44,45,46,47,48,49, 60], the Eq. (23) is

$$\frac{\left[\rho \nu {{\left(\frac{\partial {u}_1}{\partial {x}_2}\right)}^2}_{stoch}\right]\Delta {\tau}_{stoch}}{\left[\rho \nu {{\left(\frac{\partial {u}_1}{\partial {x}_2}\right)}^2}_{lamin}\right]\Delta {\tau}_l}=\frac{9}{n^2}{\left[\left({Re}\right)\left(\frac{h}{L_2}\right)\left(\frac{L_2}{x_1}\right)\right]}^{\left(2/n\right)-2/3}\frac{\Delta {\tau}_{stoch}}{\Delta {\tau}_l}.$$

(24)

Easy to see that for laminar flow n = 3, the right side of Eq. (24) is equal to 1. Let us determine the value of the left side of the Eq. (23) (or the right side of Eq. 24) for various values of the indicator “n” and then compare with the calculated value of the right side of the Eq. (23). For the left side of the equation, if n = 2.5, we have ${K}_{\tau }=\frac{\Delta {\tau}_{stoch}}{\Delta {\tau}_l}\approx 0.8\div 1.2$ [56,57,58] and in accordance with [61,62,63,64,65,66], $\left(\frac{h}{L_2}\right)\cdot \left(\frac{L_2}{x_1}\right)=\left(2.5\div 14.5\right)\cdot \left(0.01\div 0.0157\right)=0.025\div 0.22765$. As can be seen, the experimental spread of values refers to the magnitude of the turbulence scale $\left(\frac{h}{L_2}\right)$. Therefore, the main attention will be focused on the correspondence of the calculated values of the right side of Eq. (24) to the left side of this equation. Then it can be written that

$$\frac{\rho \nu {{\left(\frac{\partial {u}_1}{\partial {x}_2}\right)}^2}_{stoch}}{\rho \nu {{\left(\frac{\partial {u}_1}{\partial {x}_2}\right)}^2}_{lamin}}{K}_{\tau }=\frac{9}{6.25}\cdot {\left[\left(\frac{h}{L_2}\right)\cdot \left(\frac{L_2}{x_1}\right)\right]}^{0.1333}\cdot {\left({Re}\right)}^{0.1333}\cdot {K}_{\tau }=1.44\cdot {\left[0.025\div 0.22765\right]}^{0.1333}\cdot {\left(7.25\div 25\right)}^{0.1333}\cdot \left(0.8\div 1.2\right),$$

(25)

$$\frac{\rho \nu {{\left(\frac{\partial {u}_1}{\partial {x}_2}\right)}^2}_{stoch}}{\rho \nu {{\left(\frac{\partial {u}_1}{\partial {x}_2}\right)}^2}_{lamin }}{K}_{\tau }=(1.44)\cdot \left(0.61\div 0.821\right)\cdot {{Re}}^{0.133}\cdot \left(0.8\div 1.2\right)=\left(0.878\div 1.18\right)\cdot \left(1.3\div 1.53\right)\cdot \left(0.8\div 1.2\right).$$

(26)

So for n = 2.5, the left side of the equation is

$$\frac{\rho \nu {{\left(\frac{\partial {u}_1}{\partial {x}_2}\right)}^2}_{stoch}}{\rho \nu {{\left(\frac{\partial {u}_1}{\partial {x}_2}\right)}^2}_{lamin }}{K}_{\tau }=\left(0.913\div 1.62\right)\div \left(1.235\div 2.18\right)=1.5.$$

(27)

Then for n = 2.0 and the same initial data for the critical Reynolds number (Re)_cr ~ 7.25 ÷ 25, ${K}_{\tau }=\frac{\Delta {\tau}_{stoch}}{\Delta {\tau}_l}\approx 0.8\div 1.2$ [54,55,56,57,58] and $\left(\frac{h}{L_2}\right)\cdot \left(\frac{L_2}{x_1}\right)=\left(2.5\div 14.5\right)\cdot \left(0.01\div 0.0157\right)=0.025\div 0.22765$ [61,62,63,64,65,66,67], the next equation can be written as

$$\frac{\rho \nu {{\left(\frac{\partial {u}_1}{\partial {x}_2}\right)}^2}_{st och}}{\rho \nu {{\left(\frac{\partial {u}_1}{\partial {x}_2}\right)}^2}_{lamin }}{K}_{\tau }=\frac{9}{4}\cdot {\left[\left(\frac{h}{L_2}\right)\cdot \left(\frac{L_2}{x_1}\right)\right]}^{1/3}\cdot {\left({Re}\right)}^{1/3}\cdot {K}_{\tau }=\left|{{Re}}_{st}-\left(1/{{Re}}_{st}\right)\right|.$$

(28)

Substituting the numerical values, it is possible to find that

$$\frac{\rho \nu {{\left(\frac{\partial {u}_1}{\partial {x}_2}\right)}^2}_{stoch}}{\rho \nu {{\left(\frac{\partial {u}_1}{\partial {x}_2}\right)}^2}_{lamin }}\cdot {K}_{\tau }=\frac{9}{4}\cdot {\left[\left(\frac{h}{L_2}\right)\cdot \left(\frac{L_2}{x_1}\right)\right]}^{1/3}\cdot {\left({Re}\right)}^{1/3}\cdot {K}_{\tau }=2.25\cdot {\left[0.025\div 0.22765\right]}^{1/3}\cdot {\left(7.25\div 25\right)}^{1/3}\cdot \left(0.8\div 1.2\right).$$

(29)

So for n = 2.0, the left side of the equation is

$$\frac{\rho \nu {{\left(\frac{\partial {u}_1}{\partial {x}_2}\right)}^2}_{stoch}}{\rho \nu {{\left(\frac{\partial {u}_1}{\partial {x}_2}\right)}^2}_{lamin }}{K}_{\tau }=2.25\cdot \left(0.292\div 0.608\right)\cdot \left(0.8\div 1.2\right)\cdot \left(1.94\div 2.924\right)=\left(1.02\div 2.3\right)\div \left(2.12\div 4.81\right)\approx 2.5.$$

(30)

Then determine the right side of Eq. (23). Within the framework of the available data in the case of the origin of turbulence, experimental values of turbulent Reynolds number are Re_st ~ 0.4÷0.5 [61,62,63,64,65,66,67]. In the case of the developed turbulence, according to [2], experimental values of turbulent Reynolds number are Re_st ~ 2÷4. So the right side of Eq. (23) for n = 3.0 ÷ 2.5 has value

$$\left|{{Re}_{st}}\hbox{-} 1/{{Re}_{st}}\right|=\left|\left(0.5\div 0.4\right)\hbox{-} 1/\left(0.5\div 0.4\right)\right|\approx \left(1.5\div 2.1\right)\approx 1.7,$$

and for n = 2.0 the value is

$$\left|{{Re}_{st}}\hbox{-} 1/{{Re}_{st}}\right|=\left|\left(2\div 4\right)\hbox{-} 1/\left(2\div 4\right)\right|\approx \left(1.5\div 3.8\right)\approx 2.7.$$

(31)

Thus, the estimate of the profile index is satisfactorily determined by the obtained relation n = 2.0, corresponding to the turbulent flow value of index ‘n’ in empirical formula u₁ ~ (x₁)^-1/2 [61, 64]. It should be noted that the main last studies [61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119] do not contain information on analytical solutions for the jet profile in the laminar-turbulent transition regime.

5 Conclusions

On the basis of the theory of stochastic equations and the theory of equivalence of measures, the velocity profile of a plane jet is considered. In accordance with these theories, the analytical dependence of the velocity profile of flat jet is derived for the transition flow from the laminar flow to turbulent motion. It should be noted that the calculated values obtained from the formulas for the transition from laminar movement to turbulent flow in a plane jet are different to the calculated values for the velocity profile in the boundary layer on the flat plate and in the pipe, in the case of a smooth wall [33,34,35,36,37,38,39,40,41,42,43, 49,50,51,52]. This fact agrees with the well-known data [61, 64]. It is also seen that the analytical formulas (27) and (30) reliably reflect an increase of the transferred energy from a deterministic state to a random one with an increase of the index (1/n). It is shown that the new equation reflects the experimental fact of changing the dependence for the longitudinal velocity u₁ during laminar motion from the coordinate along the axis of the jet u₁ ~ (x₁)^-1/3 (n = 3) into a dependence for the developed turbulent region u₁ ~ (x₁)^-1/2 (n = 2). The calculations carried out using new formulas (24)–(31) showed satisfactory agreement with the known experimental values for parameters of the velocity profile of flat jet.

The practical significance of the theoretical determination of velocity profiles in a jet at a laminar-turbulent transition from the values of the Reynolds number and vice versa is essential for a number of processes taking place, in particular, in combustion chambers [30] and in the case of welding [43, 120]. Also, knowing the initial parameters of the disturbance, it is possible, depending on the distance x₁, to determine the amount of energy transferred into random motion. This makes it possible to determine the location of technical devices for reducing friction [58, 59] in the flow around aerodynamic vehicles and for maintaining the jet profile if it is necessary to ensure the stability of the flow characteristics.