Fluid Dynamics, Vol. 36, No. 3, 2001, pp. 429–436. Translated from Izvestiya Rossiiskoi Academii Nauk, Mekhanika Zhidkosti i Gaza, No. 3, 2001, pp. 102–110. Original Russian Text Copyright  2001 by Vatazhin and Ulybyshev.

Diffusion and Electrical Processes in the Turbulent Boundary Layer and in the Neighborhood of the Stagnation Point A. B. Vatazhin and K. E. Ulybyshev Received September 21, 2000

Abstract — The problem of electric current (“engine current”) formation in aircraft jet engine ducts as a result of the development of electrical diffusion boundary layers on the surfaces of the duct and internal engine components is investigated. It is assumed that the outer flow containing electrons and positive ions is quasi-neutral and that the electrical quasi-neutrality is violated (and the electric engine current develops) in the wall flow zone as a result of the difference between the electron and ion diffusion coefficients. The problem of the development of an electrical diffusion boundary layer inside the turbulent gasdynamic boundary layer on a plane surface is formulated and solved. The engine current distribution along the duct is found for various values of a turbulent viscosity on the boundary of the gasdynamic boundary layer which affect the laminar-turbulent transition point. The electrical diffusion processes that occurs when an electrically quasi-neutral hydrodynamic stream impinges on a plane surface (simulation of the flow in the neighborhood of a stagnation point) is studied. In this case the Navier-Stokes equations have a self-similar solution. It is shown that the system of electrohydrodynamic equations also has a self-similar solution. The electrical parameter fields are determined and the engine current is found on the basis of this solution.

One cause of engine charging in aircraft is the presence of charged particles in the engine duct, namely, electrons and ions formed in the combustion chamber, and their diffusion to the surfaces of internal engine components. In [1] a theory of the electrical diffusion processes taking place inside an engine was developed. The theory is based on the following physical effects. Since the ion and electron concentrations on the internal surfaces of the engine are less than those in the flow core, ions and electrons diffuse toward these surfaces (duct walls and internal elements). Since the electron diffusion coefficient is significantly greater than the ion diffusion coefficient De
Di ), the electron diffusion flow exceeds the ion flow. However, this process is limited by the electric field induced by charge separation which opposes further separation. As a result, in most of the duct ambipolar diffusion develops in a quasi-neutral medium. In this case no engine current is generated. However, in the wall flow zone whose thickness is equal to the Debye radius rd in order of magnitude there is a possibility of the formation of a space charge which is carried off by the flow from the engine into ambient space, producing the engine current I. In order to determine the total entrained current it is necessary to consider two aspects (situations). The first is the development of an electrical diffusion boundary layer on the channel walls. The second is the electrical processes in the neighborhood of the surfaces of internal engine components (turbine blades) located in an electrically quasi-neutral hydrodynamic stream containing electron and ion components. In [1] the first problem was solved in the simplified formulation in which the ion component is assumed to be “frozen” into a homogeneous hydrodynamic flow. However, under real conditions the electrical boundary layer develops inside a laminar or turbulent hydrodynamic boundary layer and these effects may significantly affect the local and integral gasdynamic characteristics. In the first stage of the investigation, the second aspect (situation) can be simulated by means of the problem of the impingement of an incompressible viscous flow transporting electron and ion components on a plane surface. 0015–4628/01/3603–0429$25.00  2001 Plenum Publishing Corporation

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In what follows, we will formulate the above two problems and analyze them theoretically and numerically. 1.

EQUATIONS OF THE ELECTRICAL DIFFUSION BOUNDARY LAYER ON A FLAT PLATE

We will consider a quasi-neutral plasma flow with ion and electron concentrations n0 past a flat plate. We will assume that the electrogasdynamic interaction parameter is small, the carrier medium is incompressible, and its temperature T is constant. In the neighborhood of the plate a laminar or turbulent boundary layer and an electrical diffusion boundary layer develop. As a rule, the thickness of the latter is less than the thickness of the hydrodynamic boundary layer [1]. In this formulation within the framework of the model [2] the dynamic equations of the turbulent boundary layer have the form: u

∂u ∂u ∂ ∂u +v = (ν + ε ) ∂x ∂y ∂y ∂y

(1.1)

∂u ∂v + =0 ∂x ∂y

(1.2)




∂ u
γε (β ε + ν ) ∂ε ∂ε ∂ ∂ε = (ν + χε ) + αε



− u +v ∂x ∂y ∂y ∂y ∂y y2  ε 0.71 ε ε α = 0.28 ,
1; α = 0.28, > 1; β = 0.34; γ = 12; χ = 2 7ν 7ν 7ν

(1.3) (1.4)

Here, x and y are the longitudinal and transverse coordinates, u and are the horizontal and vertical velocity components, ν and ε are the molecular and turbulent viscosities, and α , β , γ , and χ are empirical constants. The model [2] makes it possible to describe the laminar-turbulent boundary layer transition. The equations for the distributions of the ion and electron concentrations and the transverse electric field component in the electrical diffusion boundary layer can be written as follows: u

∂ ni ∂n ∂ Eni ∂n ∂ + v i + bi = (Di + ε ) i ∂x ∂y ∂y ∂y ∂y

(1.5)

u

∂ ne ∂ ne ∂ Ene ∂ ∂ ne +v − be = (De + ε ) ∂x ∂y ∂y ∂y ∂y

(1.6)

e ∂E = (ni − ne ); ∂y ε0

D=b

kT e

(1.7)

Here, ni and ne are the ion and electron concentrations, bi , be , Di , and De are their mobilities and the molecular diffusion coefficients, E is the transverse electric field component, e is the electron charge, ε0 is the electric constant, and k is the Boltzmann constant. The boundary conditions for solving the hydrodynamic and electric equations are the relations ne = ni = n0 , u = u0 , ε = ε (0)

x = 0,

0 < y < ∞:

y = 0,

x > 0:

ne = ni = 0, u = 0, v = 0, ε = 0

y = ∞,

x > 0:

ne = ni = n0 , u = u0 , E = 0, ε = ε (0)

(1.8)

Here, ε (0) , n0 , and u0 are the turbulent viscosity, the charge particle concentration and velocity on the edge of the boundary layer. The vanishing of the charge particle concentrations on the wall is a model boundary condition on the metal surface. The systems of equations (1.1)–(1.4) and (1.5)–(1.7) must be solved successively. FLUID DYNAMICS

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DIFFUSION AND ELECTRICAL PROCESSES IN THE TURBULENT BOUNDARY LAYER

2.

431

NUMERICAL SIMULATION OF THE ELECTRICAL DIFFUSION BOUNDARY LAYER IN A TURBULENT FLOW

The equations (1.1)–(1.8) were integrated in dimensionless variables introduced as follows:   u0 x 1 ν u0 0 0 , u = u0 u (ξ , η ), v = ξ = , η =y v (ξ , η ) rd νx 2 ξ rd ni = n0 n0i (ξ , η ), rd2 =

ne = n0 n0e (ξ , η ),

ε0 kT , e2 n0

Ed =

E = Ed E 0

(2.1) (2.2)

e r n ε0 d 0

(2.3)

Here, rd is the Debye radius and Ed is the characteristic electric field determined from the dimension rd . The electrical processes are characterized by the following dimensionless parameters: κ2 =

u0 rd , ν

λi =

2bi Ed κ, u0

λe =

2be Ed κ u0

(2.4)

The parameter κ 2 represents the Reynolds number determined on the basis of the Debye radius, and the parameters λi and λe are equal to the ratios of the drift ion and electron velocities to the free-stream velocity multiplied by 2κ. The ratios of the electron and ion diffusion coefficients to the quantity ν can be expressed in terms of the three parameters introduced above. In order to integrate the dimensionless hydrodynamic and electric equations we used a finite-difference scheme of the second order in y and the first order in x. The calculation was carried for the following parameters u0 = 100 m/s,

n0 = 108 cm−3 , −1

ν = 1.5 · 10

2

cm /s,

Di = 0.77 · 10−1 cm2 /s, −2

λi = 2.73 · 10 ;

De = 103 Di ,

λe = 27.2,

T = 1000 K,

(2.5)

κ = 38.3

We will give some of the calculation results. In Fig. 1 we have reproduced the charge particle concentration distributions for ε (0) /ν = 1. The ion profiles are more convex than the electron profiles. This indicates that a positively-charged near-wall layer is formed. At x = 1 cm the electrical boundary layer thickness is approximately equal to ten and at x = 1 m to more than 50 Debye radii. In Fig. 2 we have plotted the engine current distribution i(x) along the channel for the laminar (bold curve) and turbulent boundary layers, where ∞

i(x) =

eu(ni − ne ) dy

0

In the neighborhood of the leading edge of the plate the dynamic boundary layer is laminar and the electrical diffusion boundary layer develops in the laminar flow. In accordance with the model used for the turbulent viscosity (1.3)–(1.4), the laminar flow then becomes turbulent and the diffusion boundary layer continues to develop in the turbulent flow. With increase in the parameter ε (0) /ν the transition takes place closer to the leading edge of the plate and the i(x) obtained for the completely laminar and transition flows begin to differ at smaller x. In the case of laminar-turbulent flow the engine current is smaller than for the completely laminar flow (Fig. 2). This is associated with the fact that for the turbulent flow the electron and ion diffusion coefficient contains molecular and turbulent components, the latter being the same for both species of charged particles. The higher turbulent diffusion coefficient may smooth out the effect of the difference between the electron and ion molecular diffusion coefficients which, as shown above, also causes the onset of the electric engine current. However, as the calculations showed, the presence of turbulence FLUID DYNAMICS

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Fig. 1. Electron and positive ion concentration distributions in different cross-sections of the turbulent boundary layer. Curves 1–2, 3–4, and 5–6 correspond to the ion and electron concentration distributions at x = 1 cm, 10 cm, and 1 m

Fig. 2. Electric engine current as a function of the longitudinal coordinate for turbulent and laminar boundary layers. Curves 1–4 correspond to the turbulent viscosities in the outer flow ε (0) /ν = 0.5, 1, 3, and 7

leads to a not very large (of the order of 25%) reduction in the engine current. This is attributable to the fact that the thickness of the electrically charged near-wall zone only slightly exceeds the laminar sublayer thickness. We can confirm this conclusion by means of the following qualitative analysis. In order to estimate the parameters of the turbulent boundary layer on the flat plate we will use the following semiempirical relations [3, 4]:

δl = 30x Re−0.9 , x

δ (x) = 0.37xRe−1/5 u2∗ /u20 = 0.03 Re−1/5 x , x u y  ε (y) δ u∗ ∗ =F , Reδ , = 0.064Re0.7 x u∗ δ ν ν

(2.6)

u0 δ u x , Rex = 0 ν ν Here, δ and δl are the thicknesses of the boundary layer and its laminar sublayer, τw and u∗ are the wall friction stress and the “dynamic velocity”, Rex and Reδ are the Reynolds numbers determined from the dimensions x and δ , and ε (y) is the transverse turbulent viscosity distribution. The function F is given in [3]. For small values of the argument it can be approximated by a quadratic parabola. For large values of

τw = ρ u2∗ ,

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Reδ =

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DIFFUSION AND ELECTRICAL PROCESSES IN THE TURBULENT BOUNDARY LAYER

the argument it is worthwhile transforming the function F into a function F = Φ(y/δ , Reδ ). This function has a maximum. When Reδ = 8 · 104 this maximum is reached at the point y/δ = 0.3 and is equal to 0.085. Below, we give the results of the calculations for the boundary layer cross-sections with Rex = 106 and 107 based on the formulas (2.6) and the data of [3] for the function F: Rex

Rd , cm

Reδ

R = rd u∗ /ν

δ u∗ /ν

Φ(rd /δ , Reδ )

ε (rd )/ν

106

2.2 · 10−2

2.3 · 104

64

1.0 · 103

0.011

11

107

2.2 · 10−2

1.5 · 105

51

5.1 · 103

0.007

36

These results show that the boundary of the Debye (electrically charged) layer lies inside the completely developed turbulent layer zone in the near-wall region. However, on this boundary the turbulentto-molecular viscosity ratio is not too high. Since the laminar diffusion coefficient ratio De /Di ∼ 103
1 and Di ∼ ν , in the turbulent flow the ratio of the total electron and ion diffusion coefficients remains fairly large. 3.

ELECTRICAL DIFFUSION PROCESSES IN THE NEIGHBORHOOD OF THE STAGNATION POINT. SELF-SIMILAR EQUATIONS

We will consider the problem of the impingement of a flow of electrically quasi-neutral plasma on a body. The system of Navier-Stokes equations describing the flow in a small neighborhood of the leading stagnation point has the well-known self-similar solution [4]  √ ρα 2 x2 α + ρνα P(η ), η = y (3.1) u = α xΦ(η ), v = − αν Φ(η ), p = 2 ν Φ + ΦΦ = Φ − 1, Φ(0) = Φ (0) = 0, Φ (∞) = 1; P = −(Φ + 1/2 Φ2 ) 2

(3.2)

Here, x and y are the streamwise and transverse coordinates in the neighborhood of the stagnation point, and α = const. As η → ∞, the gas velocity components are determined from the expressions u = α x and v = −α y, which represent the solution of the equations for an inviscid medium in the neighborhood of the stagnation point. The solution of the general system of electrical equations has the structure E = (0, E(y), 0),

ne = ne (y),

ni = ni (y)

(3.3)

Substituting (3.3) in the general system of electrical equations, we obtain v

dEni d2n dni + bi = Di 2i , dy dy dy

v

dEne d 2 ne dne − be = De 2 , dy dy dy

De = be

kT e

Di = bi

(3.4)

kT e

(3.5)

e dE = (ni − ne ) dy ε0

(3.6)

The boundary conditions are formulated in the same way as in Section 2 y = 0 : ni = ne = 0,

y = ∞ : n0 = ne = n0 ,

E =0

(3.7)

The engine current can be determined from the expression ∞

I =z

e(ni − ne )u dy = xze(De ne (0) − Di ni (0))

(3.8)

0

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Fig. 3

Fig. 4

Fig. 3. Dimensionless engine current as a function of the dimensionless parameter of the problem for the boundary layer in the neighborhood of a leading stagnation point Fig. 4. Ion (curve 4) and electron (curves 1, 2, and 3) concentrations as functions of the transverse coordinate for the electrical boundary layer in the neighborhood of the stagnation point for various free-stream charged particle concentrations (n0 = 107 , 108 , and 109 cm−3 )

Here, x and z are the streamwise and perpendicular (with respect to the flow plane) dimensions of the region in the neighborhood of the stagnation point. We go over to dimensionless variables by means of the formulas ni = n0 n0i ,

ne = n0 n0e ,

y = y0 rd ,

E = Ed E 0

(3.9)

Then equations (3.4)–(3.6), the boundary conditions, and the engine current can be written in the form: dn0i dE 0 n0i d 2 n0i + = dy0 dy0 dy0 2

(3.10)

−θe Φ1 (y0 )

dn0e dE 0 n0e d 2 n0e − = dy0 dy◦ dy0 2

(3.11)

dE 0 = n0i − n0e dy0

(3.12)

   α r2 α r2 Φ1 (y0 ) = θν−1/2 Φ y0 θν , θν = d , θβ = d , β = i, e ν Dβ

(3.13)

y=0:

n0i = n0e = 0

(3.14)

y=∞:

n0i

I= FLUID DYNAMICS

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=

n0e

= 1, 

I0 =

0

E =0


dn0e Di dn0i

− dy0 De dy0
y0 =0

(3.15)

DIFFUSION AND ELECTRICAL PROCESSES IN THE TURBULENT BOUNDARY LAYER

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Fig. 5. Dimensionless electric field as a function of the transverse coordinate for the boundary layer in the neighborhood of the stagnation point at various free-stream charged particle concentrations (n0 = 107 , 108 , and 109 cm−3 )

System (3.10)–(3.14) contains three dimensionless parameters: θν , θi , and θe . Assuming that α ∼ u0 /R, where R is the characteristic radius of curvature of the body surface in the neighborhood of the stagnation point, we obtain  r 2 u R ν ν 0 d θν ∼ , θi = θν , θe = θν R ν Di De When the ratios ν /Di and ν /De are fixed (in what follows, these ratios are chosen in accordance with (2.5)), the solution depends only on a single parameter. We chose the quantity θe . In Fig. 3 we have plotted graphs of the calculated dependence of I 0 on θe . Curve 1 corresponding to the solution of Eqs. (3.10)–(3.14) differs only slightly from curve 2 corresponding to the solution of these equations obtained using with the use of the inviscid approximation for the dynamic velocity field. This is because under the conditions considered the viscous dynamic boundary layer thickness δ is much less than 8 −3 K we have rd = 2.2 · 10−2 . The thickness δ is the Debye radius. In fact, when n0 = 10 cm and T = 1000 equal to ν /α in order of magnitude, i.e., equal to 3.9 · 10−3 cm for ν = 1.5 · 10−1 cm2 /s and α = 104 s−1 . Over this range of variation of θe the dependences (1) and (2) are near-linear. Then, using (3.15), we obtain the following approximate expression for the engine current I ∼

eDe n0 xz θe ∼ en0 α rd xz rd

For flow past a circular cylinder we have α = 2u0 /R and x = ψ R, where R is the radius of the cylinder and ψ is the arc angle in the neighborhood of the stagnation point. The engine current can be determined from the following asymptotic formula: I ∼ 2u0 ψ zen0 rd . We will now give the calculated transverse distributions of the electron and ion concentrations and the electric field. In Fig. 4 we have plotted the dimensionlesselectron and ion concentrations n0e and n0i as functions of the dimensionless coordinate y/δ , where δ = ν /α , α = 104 . For different n0 the ion concentration distributions differ only slightly. In Fig. 5 we have reproduced the transverse dimensionless electric field distributions for the same concentrations n0 as in Fig. 4. A charge is formed on the surface. This charge increases with the charged particle concentration in the free stream. Summary. The process of violation of electrical quasi-neutrality and onset of an electric current (engine current) in a gas stream containing electrons and ions traveling along a plane surface or past a blunt body is studied. FLUID DYNAMICS

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The development of the electrical diffusion boundary layer both inside a hydrodynamic laminar or turbulent boundary layer and inside a boundary layer with laminar-turbulent transition is investigated. Since in the laminar boundary layer the violation of the electrical quasi-neutrality of the medium takes place as a result of a significant difference between the electron and ion molecular diffusion coefficients, it might be expected that in a turbulent flow in which the total electron and ion coefficients contain molecular and identical (large) turbulent components the violation of the electrical quasi-neutrality of the medium would be significantly reduced. However, as shown in our theoretical estimates and numerical calculations, for the engine current the reduction is not greater than 20%. This is because under real conditions the Debye wall layer thickness (in which the electric space charge is mainly concentrated) only slightly exceeds the laminar sublayer thickness. A study of the electrical diffusion boundary layer in the neighborhood of the stagnation point of a blunt body in a quasi-neutral stream shows that the complete system of electrical equations has a self-similar solution, on the basis of which local and integral electrical characteristics in the zone of violation of the quasi-neutrality of the medium can be found. In the flow considered the viscous boundary layer thickness is much less than the Debye layer thickness and, therefore, the electric engine currents calculated with allowance for the viscosity of the medium and in the inviscid approximation practically coincide. The work was supported by the Russian Foundation for Basic Research (project No. 99-01-00983). REFERENCES 1. A. B. Vatazhin and K. E. Ulybyshev “Model of electric current formation in aircraft jet engine ducts,” Izv. Ros. Akad. Nauk, Mekh. Zhidk. Gaza, No. 5, 139 (2000). 2. E. K. Kholshchevnikova, “Study of the possibility of using a differential model of the turbulent viscosity for calculating various flows in a boundary layer,” Tr. TsIAM, No. 994 (1982). 3. J. O. Hinze, Turbulence. McGraw-Hill, New York (1959). 4. H. Schlichting, Boundary Layer Theory, McGraw-Hill, New York (1968).

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