On the Breaking of Mirror Symmetry in Homogeneous Isotropic Turbulence-Helicity Effect

H.P.Mazumdar, Gr.Tsagas and S.C.Ghosh

Dedicated to Prof.Dr. Constantin UDRIŞTE on the occasion of his sixtieth birthday

Abstract

In homogeneous isotropic turbulence when reflexional (or mirror) symmetry breaks down, it becomes necessary to take accounts of the additional fundamental invariants in constructing forms for correlation tensors of different orders that characterize the flow. It is helicity which plays its role in such a situation. Attention is paid here, in particular to the construction of appropriate invariant forms for the several two-point second and third order correlation tensors that arise out of fluctuating pressure, velocity and external force fields of such a turbulent flow. Utilizing these forms of second and third order tensors, the two-point vorticity-vorticity and acceleration-acceleration correlations are determined, separately illustrating the effects of helicity on them.

Mathematics Subject Classification: 76F05, 76D05 Key words: isotropic turbulence, correlation tensors, symmetry, vorticity, Navier-Stokes equation

1 Introduction

The problems of turbulent flows are studied much in some simplifying situations as to make their mathematical representation possible and to verify the results experimentally. The turbulence is said to be simply homogeneous if the statistical parameters describing the flow, remain unchanged under the translation of the axes. The turbulence is said to be isotropic, or homogenous isotropic if the statistical parameters e.g., correlation functions of different orders remain invariant with respect to entire rotation group of transformations and as well as reflexion at an arbitrary point. The mean velocity is zero in homogeneous isotropic turbulence. It is well known that such turbulence can approximately be produced in the laboratory. Kitt et al. [1,2] however, demonstrated experimentally that grid generated homogeneous turbulence (invariant under the rotation group of transformations) may lack reflexional (or mirror) symmetry and possess significant mean helicity < u⃗ · ϖ⃗ >, where u⃗ and ϖ⃗ are, respectively the velocity and vorticity fields and < · > indicates the ensemble average.

Balkan Journal of Geometry and Its Applications, Vol.5, No.1, 2000, pp. 119-128 \begin{matrix}c \\ ◯ \end{matrix}Balkan Society of Geometers, Geometry Balkan Press

Our aim in this paper, is to examine the constructions of tensorial forms for twopoint correlation functions of different orders and their applications to fluctuating vorticity-vorticity and acceleration-acceleration correlations in homogeneous isotropic turbulence without possessing reflexional (or mirror) symmetry.

The fluid will be assumed incommpressible throughout the analysis (that is, density is constant).

2 Correlation tensors and their forms

Various quantities, namely the velocity, the vorticity, the acceleration, the pressure, the temperature or the density fluctuations etc., which characterize the turbulent flow are considered as random functions in the four dimensional space (r⃗ , t) (Lesieur [3]). In homogeneous isotropic turbulence possessing reflexional symmetry, the correlation functions of different orders formed from these quantities remain invariant, in geometrical sense, under arbitrary rigid rotations and reflexions of the configuration of the vector arguments. For instance, in respect of two arbitrary fixed vectors a⃗ and b⃗, the tensorial product a_{i}U_{ij}(r⃗ , t′t′′)b_{j}, where U_{ij}(r⃗ , t′, t′′) =< u′_{i}(r⃗ ′, t′)u′_{j}(r⃗ ′′, t′′) >, r⃗ ′′ − r⃗ ′ = r⃗ , is a scalar and accordingly, must be invariant under translation and rotation of the three vectors (r⃗ , a⃗, b⃗) · u′_{i}, u′′_{j} are the fluctuating velocity components. Robertson [4] (see also Batchelor [5], Hinze [6]) showed that such an invariant function can be expressed in terms of the fundamental invariants due to a rigorous result of group invariant theory. The invariants are in this case the scalar products r⃗ · r⃗ , r⃗ · a⃗, r⃗ · b⃗, a⃗ · a⃗, b⃗ · b⃗ and a⃗ · b⃗. If the reflexional (or mirror) symmetry breaks down, an additional fundamental invariant r⃗ · (a⃗ × b⃗) is to be taken into account. In this last case helicity of the flow plays an important role, which will be illustrated in sections 3 and 4.

We now describe the forms for the correlation tensors of first, second and third orders pertaining to two points and at two different times in a non-stationary homogeneous isotropic turbulence possessing reflexional symmetry.

Homogeneous isotropic turbulence possessing reflexional symmetry

(1)  L_{i}(r⃗ , t′, t′′) =< θ′(r⃗′, t′)u_{i}(r⃗′′, t′′) >= Lr_{i},

r⃗′′ − r⃗′ = r⃗ , θ′ is a scalar, namely fluctuating pressure, temperature or density.

(2)  U_{ij}(r⃗ , t′, t′′) =< u′_{i}(r⃗′, t′)u′′_{j}(r⃗′′, t′′) >= Aδ_{ij} + Br_{i}r_{j}

(3)  \begin{aligned}T_{ij;k}(r⃗ , t′, t′′) & = & < u′_{i}(r⃗′, t′)u′_{j}(r⃗′, t′)u′′_{k}(r⃗′′, t′′) >= \\ & = & T_{1}r_{i}r_{j}r_{k} + T_{2}r_{i}δ_{jk} + T_{3}r_{j}δ_{ki} + T_{4}r_{k}δ_{ij} \end{aligned}

(4)  A_{ij}(r⃗ , t′, t′′) =< A′_{i}(r⃗′, t′)A′′_{j}(r⃗′′, t′′) >= aδ_{ij} + βr_{i}r_{j},

where A′_{i} and A′′_{j} are the fluctuating components of acceleration.

(5)  P_{ij}(r⃗ , t′, t′′) =< u′_{i}(r⃗′, t′)u′_{j}(r⃗′, t′)\frac{p′′}{ρ} (r⃗′′, t′′) >= P_{1}r_{i}r_{j} + P_{2}δ_{ij},

where p′′ is the pressure fluctuating and ρ is the density of the fluid.

(6)  ϖ_{ij}(r⃗ , t′, t′′) =< ϖ′_{i}(r⃗′, t′)ϖ′′_{j}(r⃗′′, t′′) >= a^{∗}δ_{ij} + β^{∗}r_{i}r_{j},

where ϖ′_{i} and ϖ′′_{j} are the fluctuating components of vorticity. We can write forms similar to (1) for pressure-velocity correlation tensors like

B_{i} =< u′_{i}(r⃗′, t′)p′′(r⃗′′, t′′) >

and

B_{j} =< p′_{i}(r⃗′, t′)u′′_{j}(r⃗′′, t′′) > .

The defining scalars in (1)-(6), namely L, A, B, T_{1}, T_{2}, T_{3}, T_{4}, a, β, P_{1}, P_{2}, a^{∗}, β^{∗} are all functions of r = |r⃗′′ − r⃗′|, t′ and t′′. But in the case when the turbulence is homogeneous isotropic and stationary, all these defining become functions of r and t = |t′′−t′|. It is to be mentioned that for a such a stationary turbulence a forcing term is appended in the dynamical equation (Navier-Stokes) for sustaining the turbulence.

Form for the fourth order correlation tensor of fluctuating velocity components, namely

U_{ij;kl} =< u′_{i}(r⃗′, t′)u′_{j}(r⃗′, t′)u′′_{k}(r⃗′′, t′′)u′′_{l}(r⃗′′, t′′) >

can be obtained easily by Robertson’s invariant theory (see Batchelor [5]). The incom-

pressibility condition \frac{∂u_{i}}{∂r_{i}} = 0, \frac{∂}{∂r_{i}} U_{ij} = 0 or the solenoidal condition \frac{∂}{∂r_{k}} T_{ij;k} = 0

will help in reducing the number of defining scalars. It can be shown easily that all first order tensors like correlation between a scalar (pressure, temperature etc.) and a vector, vanish in homogeneous isotropic turbulence whether reflexional symmetry holds or not (see Lesieur [3]).

Homogeneous isotropic turbulence without possessing reflexional symmetry

In this case,

(7)  U_{ij} = Aδ_{ij} + Br_{i}r_{j} + Cϵ_{ijs}r_{s},

where C is an additional defining scalar. Henceforth, the dependence of the defining scalars will be assumed on the absolute space separation and on absolute time separation or different times, accordingly as the turbulence is stationary or non-stationary.

We now choose the frame of reference as such r⃗ has components (r, 0, 0). Therefore,

(8)  \begin{matrix}U_{11} = A + r^{2}B = F, & say \\ U_{22} = A = G, & say \\ U_{23} = rC = H, & say \end{matrix} \} .

Due to incompressibility condition \frac{∂}{∂r_{i}} U_{ij} = 0 and it can be easily shown that

(9)  G = F + \frac{r}{2} \frac{∂F}{∂r} .

In view of the relations (8) and (9), the relation (7) is transformed to

(10)  U_{ij} = (F + \frac{rF}{2} ) δ_{ij} + (−\frac{1}{2r} F ) r_{i}r_{j} + \frac{H}{r} ϵ_{ijs}r_{s}.

The mean helicity is defined by H_{e} = \frac{1}{2} < u⃗(r⃗ , t) · [∇ × u⃗(r⃗ , t)] > and it can be shown that H_{e} = −3C(0, t) (see Lesieur [3]).

(11)  \begin{aligned}T_{ij;k} & = & −\frac{T′}{r} r_{i}r_{j}r_{k} − \frac{1}{2} (rT′ + 3T) (r_{i}δ_{jk} + r_{j}δ_{ki}) + T r_{k}δ_{ij}+ \\ & + & M (ϵ_{jkm}r_{m}r_{i} − ϵ_{kim}r_{m}r_{j}) . \end{aligned}

It is to be noticed in the form (11) of the triple-velocity correlation that a single defining scalar T has appeared due to solenoidal condition at the index k and M has appeared due to lack of reflexional symmetry. Also, as u′_{i} and u′_{j} being both at the same point, T_{ij;k} should not be changed if i and j are interchanged. The form (11) clearly satisfies this condition.

(12)  A_{ij} = aδ_{ij} + βr_{i}r_{j} + γϵ_{ijs}r_{s}.

As reflexional symmetry does not hold, the third term with defining scalar γ has appeared in the form (12) of correlation between acceleration fluctuations.

(13)  P_{ij} = P_{1}r_{i}r_{j} + P_{2}δ_{ij}.

P_{ij} being the second order correlation < u′_{i}u′_{j} \frac{p′′}{ρ} >, it will remain unchanged if i

and j are interchanged. So, lack of reflexional symmetry does not affect the form of P_{ij}.

The correlation between the vorticity fluctuations ϖ_{ij} will have the form

(14)  ϖ_{ij} = a^{∗}δ_{ij} + β^{∗}r_{i}r_{j} + γ^{∗}ϵ_{ijs}r_{s}.

γ^{∗} is the additional defining scalar appeared due to same reason as in (12).

In the case of homogeneous isotropic stationary turbulence we shall have to con-

sider external forcing term f′_{i}, which is assumed to be divergence free i.e., \frac{∂f′_{i}}{∂r′_{i}} = 0.

Accordingly, we need to consider the following correlation functions

(15)  M_{ij} =< u′_{i}f′′_{j} >= (M_{1} + \frac{rM′_{1}}{2} ) δ_{ij} + (−\frac{M′_{1}}{2r} ) r_{i}r_{j} + \frac{M_{2}}{r} ϵ_{ijs}r_{s}

(16)  G_{ij} =< f′_{i}u′′_{j} >= (G_{1} + \frac{rG′_{1}}{2} ) δ_{ij} + (−\frac{G′_{1}}{2r} ) r_{i}r_{j} + \frac{G_{2}}{r} ϵ_{ijs}r_{s}

(17)  S_{ij} =< f′_{i}f′′_{j} >= (S_{1} + \frac{rS′_{1}}{2} ) δ_{ij} + (−\frac{S′_{1}}{2r} ) r_{i}r_{j} + \frac{S_{2}}{r} ϵ_{ijs}r_{s}

(18) \begin{aligned}I_{ij;k} & = & < u′_{i}u′_{j}f′′_{k} >= −\frac{I′}{r} r_{i}r_{j}r_{k}− \\ & − & \frac{1}{2} (rI′ + 3I) (r_{i}δ_{jk} + r_{j}δ_{ki}) + Ir_{k}δ_{ij} + J (ϵ_{jkm}r_{m}r_{i} − ϵ_{kim}r_{m}r_{j}) . \end{aligned}

The forms (15), (16) and (17), respectively for M_{ij}, G_{ij} and S_{ij} are constructed in the same manner as in (10) for U_{ij}. M_{2}, G_{2} and S_{2} are the defining scalars due to lack of reflexional symmetry. The form for the triple order correlation I_{ij;k} involves single defining scalar I similar to T in the form of T_{ij;k} in (11). The defining scalar J has appeared in (18) due to the violation of reflexional symmetry.

It is to be noted that the defining scalars F in (10), T in (11), M_{1} in (15), G_{1} in (16) and S_{1} in (17) with primes indicate their differentiations with respect to r.

3 Correlation between components of vorticity at two different points

The turbulent vorticity ϖ⃗(r⃗′, t) at a point (r⃗′, t), say is defined by

(19)  \overrightarrow{ϖ} (r⃗′, t) = ∇ × u⃗ (r⃗′, t) .

The components of vorticity is given by

(20)  ϖ_{i} (r⃗′, t) = ϵ_{ijk} \frac{∂u_{k} (r⃗′, t)}{∂r_{j}};

ϖ_{1}, ϖ_{2} and ϖ_{3} are the components of vorticity.

The correlation tensor formed from the components of vorticity at two different points r⃗ ′ and r⃗ ′′ and at the same instant of time t, say can be expressed from kinematical consideration as (Batchelor [5])

(21)  \begin{aligned}ϖ_{ij}(r⃗ , t) & = & < ϖ′_{i}(r⃗′, t)ϖ′′_{j}(r⃗′′, t) >= \\ & = & ϵ_{ilm}ϵ_{jpg} \frac{∂u′_{m}(r⃗′, t)}{∂r′_{l}} \frac{∂u′′_{q}(r⃗′′, t)}{∂r′′_{p}}− \\ & = & −δ_{ij}∇^{2}U_{ll} (r⃗ , t) + \frac{∂^{2}U_{ll} (r⃗ , t)}{∂r_{i}∂r_{j}} + ∇^{2}U_{ji} (r⃗ , t) , \end{aligned}

where r⃗ = |r⃗′′ − r⃗′|; r_{i}(i = 1, 2, 3) are the components of r⃗.

Now, we assume that the turbulence is homogeneous isotropic but lacks reflexional symmetry. The expression (10) for U_{ij} and the expression (14) for ϖ_{ij} are recalled here again, as

(22)  U_{ij} (r⃗ , t) = (F + \frac{rF′}{2} ) δ_{ij} + (−\frac{1}{2r} F′) r_{i}r_{j} + \frac{H}{r} ϵ_{ijs}r_{s}

(23)  ϖ_{ij} (r⃗ , t) = a^{∗}δ_{ij} + β^{∗}r_{i}r_{j} + γ^{∗}ϵ_{ijs}r_{s};

F, H, a^{∗}, β^{∗} and γ^{∗} are functions of r (= \left|{ \atop r}\right|) and t.

Taking (22) into account on the right hand side of (21) we obtain, after a long but straight forward calculations (24)

\begin{aligned}ϖ_{ij}(r⃗ , t) & = & [−\frac{r}{2}F′′′ − 3F′′ − \frac{2}{r}F′] δ_{ij} + [\frac{1}{2r} F′′′ + \frac{7}{2r^{2}} F′′ − \frac{7}{2r^{3}} F′] r_{i}r_{j}+ \\ & + & [−\frac{H′′}{r} − \frac{2H′}{r^{2}} + \frac{2H}{r^{3}} ] ϵ_{ijs}r_{s}. \end{aligned}

Comparing (23) with (24), we obtain

(25)  \begin{matrix}a^{∗} = −\frac{r}{2}F′′′ − 3F′′ − \frac{2}{r}F′ \\ β^{∗} = −\frac{r}{2r} F′′′ + \frac{7}{2r^{2}} F′′ − \frac{7}{2r^{3}} F′ \\ γ^{∗} = −\frac{H′′}{r} − \frac{2H′}{r^{2}} + \frac{2H}{r^{3}} . \end{matrix}

It is to be noted that F′′′, F′′ and F′ are, respectively the triple, double and single derivatives of F with respect to r. Similarly, H′′ and H′ are the double and single derivative of H with respect to r.

Since, the frame of reference be as such r⃗ has the components (r, 0, 0). That is, r_{1} = r, r_{2} = 0, r_{3} = 0. We therefore, obtain from (24) easily

(26)  \begin{matrix}< ϖ′_{1}ϖ′′_{1} >= a^{∗} + β^{∗}r^{2}, < ϖ′_{2}ϖ′′_{2} >=< ϖ′_{3}ϖ′′_{3} >= a^{∗} \\ < ϖ′_{1}ϖ′′_{3} >=< ϖ′_{1}ϖ′′_{2} >= 0 \\ < ϖ′_{2}ϖ′′_{3} >= γ^{∗}r. \end{matrix}

From (25) and (26), it follows that helicity < u′_{2}u′′_{3} > (= H) of the flow is related to cross correlation of vorticity fluctuations < ϖ′_{2}ϖ′′_{3} > in homogeneous isotropic turbulence without possessing reflexional (mirror) symmetry.

4 Correlation between fluctuations of total acceleration at two different points

In this case, we assume the turbulence to be homogeneous isotropic, stationary and without possessing reflexional symmetry.

The Navier-Stokes equation for u′_{i}(r⃗′, t′) with external force term f′_{i}(r⃗′, t) at point (r⃗′, t′) is written as

(27)  \frac{∂u′_{i}}{∂t′} + \frac{∂}{∂r′_{j}} u′_{i}u′_{j} = −\frac{1}{ρ} \frac{∂p′}{∂r′_{i}} + ν∇_{r′}^{2}u′_{i} + f′_{i},

where p′(r⃗′, t′) is the pressure fluctuation.

The total acceleration fluctuation A′_{i}(r⃗′, t′) is given by

(28)  A′_{i}(r⃗′, t) = \frac{Du′_{i}}{Dt′} = \frac{∂u′_{i}}{∂t′} + u_{k} \frac{∂u′_{i}}{∂r′_{k}} .

Taking (28) into account in (27), we obtain

(29)  A′_{i} = −\frac{1}{ρ} \frac{∂p′}{∂r′_{i}} + ν∇_{r}^{2}u′_{i} + f′_{i}.

In a similar manner, we can write for acceleration fluctuation A′′_{j}(r⃗′′, t′′) at the point (r⃗′′, t′′) as

(30)  A′′_{j} = −\frac{1}{ρ} \frac{∂p′′}{∂r′′_{j}} + ν∇_{r′′}^{2}u′′_{j} + f′′_{j}.

Now, multiplying (30) with (29) and then averaging the resultant equation, we obtain

(31)  \begin{aligned}< A′_{i}A′′_{j} > & = & \frac{1}{ρ^{2}} \frac{∂^{2}}{∂r′_{i}∂r′′_{j}} < p′p′′ > +ν^{2}∇_{r′′}^{2}∇_{r′}^{2} < u′_{i}u′′_{j} > \\ & + & ν∇_{r′′}^{2} < f′_{i}u′′_{j} > +ν∇_{r′}^{2} < u′_{i}f′′_{j} > + < f′_{i}f′′_{j} > . \end{aligned}

In view of the definitions of the correlations involved, we can rewrite equation (31) as

(32)  A_{ij} = −\frac{1}{ρ^{2}} \frac{∂^{2}}{∂r_{i}∂r_{j}} < p′p′′ > +ν^{2}∇_{r}^{4}U_{ij} + ν∇^{2}(G_{ij} + M_{ij}) + S_{ij},

since

r′′_{i} − r′_{i} = r_{i}, \frac{∂}{∂r′′_{i}} = −\frac{∂}{∂r′_{i}} = \frac{∂}{∂r_{i}} , ∇_{r′′}^{2} = ∇_{r′}^{2} = ∇_{r}^{2} = \frac{∂^{2}}{∂r_{i}∂r_{i}} and |r⃗ | = r.

Applying (12), (15), (16) and (17) in equation (32), we obtain

(33)  \begin{matrix}aδ_{ij} + βr_{i}r_{j} + γϵ_{ijs}r_{s} = −\frac{1}{ρ^{2}} \{[\frac{1}{r} \frac{∂}{∂r} (\frac{P}{r} )] r_{i}r_{j} + \frac{P′}{r} δ_{ij}\} − \\ −\frac{ν^{2}}{2} (\frac{F^{(v)}}{r} + \frac{8F^{(iv)}}{r^{2}} − \frac{24F′′}{r^{4}} + \frac{24F′}{r^{5}} ) r_{i}r_{j}− \\ −\frac{ν^{2}}{2} (−rF^{(v)} − 10F^{(iv)} − \frac{16F′′′}{r} + \frac{8F′′}{r^{2}} − \frac{8F′}{r^{3}} ) δ_{ij}+ \\ +ν^{2} (\frac{H^{(iv)}}{r} + \frac{4H′′′}{r^{2}} − \frac{4H′′}{r^{3}} ) ϵ_{ijs}r_{s}+ \\ +ν [−\frac{1}{2} (\frac{1}{r}(G′′′_{1} + M′′′_{1}) + \frac{4}{r^{2}} (G′′_{1} + M′′_{1}) − \frac{4}{r^{3}} (G′_{1} + M′_{1})] r_{i}r_{j}+ \\ +ν [−\frac{1}{2}(r((G′′′_{1} + M′′′_{1}) + 2(G′′_{1} + M′′_{1}) − \frac{12}{r}(G′_{1} + M′_{1})] δ_{ij}+ \\ +ν [D_{5}(G_{3} + M_{3})] ϵ_{ijs}r_{s} − \frac{1}{2r} S′_{1}r_{i}r_{j} + (S_{1} + \frac{r}{2}S′_{1}) δ_{ij} + S_{3}ϵ_{ijs}r_{s}, \end{matrix}

where

P =< p′p′′ >, G_{3} = \frac{G_{2}}{r} , M_{3} = \frac{M_{2}}{r} , S_{3} = \frac{S_{2}}{r} and D_{5} = \frac{∂^{2}}{∂r^{2}} + \frac{4}{r} \frac{∂}{∂r} .

The quantities F^{(v)}, F^{(iv)} represent respectively fifth and fourth order derivatives of F with respect to r. H^{(iv)} and H′′′ are respectively the fourth order and third order derivatives of H with respect to r. The quantities Gs, Ms and Ss with single, double and triple primes represent, respectively their first, second and third order derivatives with respect to r.

It is to be mentioned here that all the defining scalars appeared in (33) are functions of r and t = |t′′ − t′|, since the turbulence is homogeneous isotropic and stationary. Ghosh [7], after a long calculation derived the relation between P =< p′p′′ > and F as

(34)  \frac{1}{ρ^{2}} \frac{∂}{∂r} (\frac{P′}{r} ) = \frac{4F^{′2}}{r} .

For convenience of the readers, we state, briefly the procedure of deriving equation (34).

Multiplying equation (27) by p′′ at (r⃗′′, t′′) and then averaging the resultant equation, one obtains

(35)  \frac{1}{ρ^{2}} \frac{∂}{∂r′_{i}} < p′p′′ >= −\frac{∂}{∂r′_{j}} < u′_{i}u′_{j} \frac{p′′}{ρ} >

since < u′_{i}p′′ >= 0 and < f′_{i}p′′ >= 0, as the first order solenoidal tensors are identically zero also in homogeneous isotropic turbulence without possessing reflexional symmetry.

From (35), with the help of (13), it can be easily derived

(36)  \frac{1}{ρ^{2}} \frac{∂}{∂r} < p′p′′ >= −(r^{2}P′_{1} + 4rP_{1} + P_{2}).

Considering the equation for u′′_{i} at (r⃗′′, t′′), namely

(37)  \frac{∂u′′_{i}}{∂t′′} + \frac{∂}{∂r′′_{k}} u′′_{i}u′′_{j} = −\frac{1}{ρ} \frac{∂p′′}{∂r′′_{i}} + ν∇_{r′}^{2}u′′_{i} + f′′_{i}

and multiplying it by u′_{j}u′_{l} and then finally averaging, an equation connecting third order correlation tensor to fourth order correlation tensor is obtained as

(38)  \frac{∂}{∂t}t_{jl;i} + ν∇_{r}^{2}T_{jl;i} + I_{jl;i} = \frac{∂}{∂r_{k}} Q_{jl;ik} + \frac{∂}{∂r_{i}} P_{jl},

where Q_{jl;ik} =< u′_{j}u′_{l}u′′_{i}u′′_{k} > .

One may write

(39)  X_{jl;i} = \frac{∂}{∂r_{k}} Q_{jl;ik} + \frac{∂}{∂r_{i}} P_{jl}.

Now, applying the quasi-normally hypothesis that the fourth order moment is related to second order moments as in a normal distribution (Chandrasekhar [8]), namely

(40)  \begin{aligned}< u′_{j}u′_{l}u′′_{i}u′′_{k} > & = & < u′_{j}u′_{l} >< u′′_{i}u′′_{k} > + < u′′_{i}u′_{l} >< u′_{j}u′′_{k} > + \\ & + & < u′′_{i}u′_{j} >< u′′_{k}u′_{l} >, \end{aligned}

the solenoidality of X_{jl;i} in the index i and X_{ll;i} = 0 to equation (39), the following two relations can be obtained:

(41)  P′′_{1} + \frac{6P′_{1}}{r} = −\frac{F′F′′}{r} − \frac{6F^{′2}}{r^{2}} − \frac{HH′}{r^{3}} + \frac{4H^{2}}{r^{4}}

(42)  P′_{2} − \frac{P_{2}}{r} − rP′_{1} = rF′F′′ + 2F^{′2} + \frac{4HH′}{r} − \frac{4H^{2}}{r^{2}} .

Eliminating P_{1} and P_{2} from (41) and (42), equation (34) is obtained.

It is to mentioned that forms for the third order tensors T_{ij;k} and I_{ij;k} given, respectively by (11) and (18) are used in deriving equation (34).

Taking (34) into consideration and comparing coefficients to both sides of (33), we finally obtain: (43)

\begin{aligned}a^{∗}(r, t) & = & − ∫ \frac{4F^{2}}{r} dr − \frac{ν^{2}}{2} (−rF^{(v)} − 10F^{(iv)} − \frac{16F′′′}{r} + \frac{8F′′}{r^{2}} − \frac{8F′}{r^{3}} ) + \\ & + & \frac{ν}{2} [r(G′′′_{1} + M′′′_{1}) + 2(G′′_{1} + M′′_{1}) − \frac{12}{r}(G′_{1} + M′_{1})] + (S_{1} + \frac{r}{2}S′_{1}) \end{aligned}

(44)  \begin{aligned}β^{∗}(r, t) & = & −\frac{4F^{′2}}{r^{2}} − \frac{ν^{2}}{2} (\frac{F^{(v)}}{r} + \frac{8F^{(iv)}}{r^{2}} − \frac{24F′′}{r^{4}} + \frac{24F}{r^{5}} ) − \\ & − & \frac{ν}{2} [\frac{1}{r}(G′′′_{1} + M′′′_{1})) + \frac{4}{r^{2}} (G′′_{1} + M′′_{1}) − \frac{4}{r^{3}} (G′_{1} + M′_{1}) − \frac{S′_{1}}{2r} \end{aligned}

(45)  γ^{∗}(r, t) = ν^{2} (\frac{H^{(iv)}}{r} + \frac{4H′′′}{r^{2}} − \frac{4H′′}{r^{3}} ) + ν[D_{5}(G_{3} + M − 3)] + S_{3}.

Thus to determine the defining scalars α^{∗}, β^{∗} and γ^{∗} of the second order accelaration correlation, we need to know the defining scalars F s, Ms, Gs, Ss of the respective second order correlations (10), (15), (16), (17) and H, involved in all these correlations. Remark 1. Turbulent flow is essentially rotational in nature, so the result that the helicity < u′_{2}u′′_{3} > (= H) is closely related to the second order cross correlation of vorticity fluctuations, namely < ϖ′_{2}, ϖ′′_{3} > obtained in the present calculation is important. Remark 2. Determination of second order correlation of acceleration fluctuations is more complicated that the determination of correlation of vorticity fluctuations in homogeneous isotropic turbulence without possessing reflexional (or mirror) symmetry. Remark 3. The present calculation warrants extensive measurements of correlation functions in the turbulent flow under consideration.

References

[1] E.Kitt, E. et al., Phys. Fluids 30, p.3323 (1987).

[2] E.Kitt, E. et al., Fluid Dyn. Res. 3, p.289 (1988).

[3] M.Lesieur, Turbulence in Fluids, Kluwer, Dordrecht, edition 1998.

[4] H.P.Robertson, Proc. Cambridge Phill. Soc. 36, p.209 (1940).

[5] G.K.Batchelor, The Theory of Homogeneous Turbulence, Cambridge University Press, Cambridge, edition 1970.

[6] J.O. Hinze, Turbulence, McGraw-Hill, New York, 1975.

[7] S.Ghosh, PhD. Thesis, Jadavpur University, Calcultta, 1996.

[8] S.Chandrasekhar, Proc.Roy. Soc. 233A, p.322 (1955).

H.P. Mazumdar On leave of absence from Physics and Applied Mathematics Unit Indian Statistical Institute Calcutta 700035, India. hpm@isical.ac.in

Gr.Tsagas Department of Mathematics and Physics Aristotle University of Thessaloniki Thessaloniki 54006, Greece.

S.C.Ghosh Society for Theoretical and Applied Fluid Mechanics Calcuta 700056, India.