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Brazilian Journal of Chemical Engineering
Print version ISSN 01046632Online version ISSN 16784383
Braz. J. Chem. Eng. vol.17 n.47 São Paulo Dec. 2000
http://dx.doi.org/10.1590/S010466322000000400006
THE INCLUSION OF SURFACE CHARACTERISTICS IN EQUATIONS FOR TURBULENCE CLOSURE
A.S.Telles
School of Chemistry, Universidade Federal do Rio de Janeiro, UFRJ, Centro de Tecnologia,
Bloco E. Rio de Janeiro  RJ, Brazil
Email affonso@h2o.eq.ufrj.br
(Received: September 20, 1999 ; Accepted: April 6, 2000)
Abstract  The possibility of the inclusion of the charactheristics of surfaces in the formulation of closure equations for the turbulent fluxes is considered. The general formulation is presented in the form of an expression which, with respect to the fluid motion, it is a general constitutive equation as proposed by Noll, and to which, it was added the explicit dependence upon the wall configuration. Restrictions imposed by frame indifference, and by the symmetry group of the surface are examined.
Keywords: closure equations, turbulence closure, inclusion of the charactheristics of surfaces
INTRODUCTION
Turbulent fluxes, and higher order correlations in turbulent flows, require the establishment of "closure equations". We propose to view these not as constitutive equations analogous those representing physical properties, but as properties of the variety of turbulent flows. As such, they should include geometric parameters describing the shape of the surfaces about which the flow is taking place.
Flow inhomogeneities, and the damping of the normal component of the fluctuating velocity are direct consequences of the presence of walls. Wall curvatures, convexity, or concavity alters the flow properties. The presence of singularities such as edges, corners, or steps introduce effects that must be taken into account. If these are relevant factors, then it should be better to have the surface properties included in the closure equations, than to have different equations for different wall configurations.
In examining the consequences of frame indifference, the change of variables accompanying a change of frame must be applied simultaneously to the motion of the fluid and to the walls. If the dependence upon the wall configuration is not made explicit on the closure equations, there result isotropic algebraic models, which are rigorously valid only for flows around a spherical surface. Flows in the neighborhood of all other surfaces require adjustment such as those introduced in the form of "wall corrections".
Questions on the applicability of material frame indifference, or of isotropy restrictions are relevant, and of great import. Frame indifference has been justly questioned since the work of Müller (1972), initiating a long discussion on the literature, recently reviewed by Speziale (1998). Murdoch (1983) presented arguments justifying the inclusion of the spin in the list of state variables, in violation of the principle of frame indifference. His arguments were expanded by Speziale (1998) who demonstrated a proposition, based upon the invariance properties of Boltzman equation assumed to be the molecular basis for the macroscopic constitutive equations. Speziale considered, in addition, the invariance properties of the transport equation for the fluctuating component of the velocity in turbulent flows. In both these instances, it is concluded that closure equations are not objective, and frame dependence is introduced by the spin. Thus, Murdoch hypothesis is validated and the list of state variables should contain the intrinsic spin tensor, defined as the combination of the spin in the motion plus the spin of the frame with respect to an inertial frame. Speziale further demonstrated that there is a strong separation of microscopic to macroscopic time scales in most continuum flows, and dependence in the spin can be neglected in constitutive equations defining physical properties of simple materials. Thus, to a high order of precision the principle of frame indifference applies. On the other hand, the ratio of time scales in turbulent fluctuations to the time scale of the average flow is not necessarily small and the closure equations of turbulence may depend, significantly upon the intrinsic spin. An opposing view is taken by Wang (1997), in a recent paper based on the full group of rigid body transformations. This was refuted by Spalart and Speziale (1999), clearly in favor of the inclusion of the spin.
In the present work, it is assumed that the closure equations for the properties of turbulent flows depend explicitly upon the solid surfaces about which the flow is taking place. Surface parameters as the unit normal, and distances from the wall have been employed with some frequency. This practice has been criticized on the basis that these cannot be uniquely defined for complex surfaces. The criticism applies for nonsmooth surfaces for which the differential surface parameters are not defined. However, for almost everywheresmooth surfaces the unit normal, the metric defining the arc length and the two main curvatures are valid surface characteristics. The fact that these are variables along the surface implies the necessity of a functional dependence on the surface parameters, "integrated" over the whole surface.
SURFACES AND CHANGES OF FRAME
Consider the turbulent flow of a Newtonian and incompressible fluid in a region of the Euclidean space, in presence of walls. Walls are twodimensional surfaces defined by a set of almost everywheresmooth mappings from an open set A of into the Euclidean space.. is a generic point in the surface, and is the surface. A moving surface is a time dependent family of surfaces . The relative description of the surface’s motion is obtained by eliminating x between the function written for two different instants of time: . In the case of rigid surfaces, the motion is given by , and its relative description assumes the form , where is an orthogonal tensor function of time, and is a moving point in space. From this expression, it is possible to calculate the velocity of a point in the surface taking the derivative with respect to and letting ® t. = W_{s} (s  c) + where is the surface spin.
A change of frame corresponds to a rigid, time dependent transformation of space into itself. If x is a point in space, then its position in the new frame will be . Truesdell and Noll (1965) employed the full group, containing arbitrarily accelerating frames. It allows all rigid displacements including time dependent rotations and linear accelerations. Müller (1972) employs the Galilean group which requires nonrotating frames and constant linear velocity, while Speziale (1998) justifies the use of arbitrary linear accelerations, but limits the transformation to nonrotating frames. His arguments are based upon Einstein principle of frame equivalence, which allows linear accelerations. In this work, the proposal of Speziale will be followed.
A motion for the fluid is a time dependent displacement from positions x at time t to position y at a different time . This is given by: . Under a change of frame the fluid motion and the surface transform as points of space:

(1) 
(2) 
If the wall configuration is altered the flow field in its neighborhood is accordingly altered and the closure equations as response functions are altered. It is therefore necessary to include the wall configuration as a variable in the closure equations, and consequently the following is posed:

(3) 
In this equation, H is any one of the correlations appearing in the transport equations for turbulent fields, for which closure equations must be prescribed. It is assumed to be an objective property, either a scalar, or a vector or a second order tensor. With respect to the fluid motion, equation (3) is a general constitutive equation as proposed by Noll [see Truesdell and Noll (1965)] expressing the functional dependence upon the fluid displacement to the present position from positions occupied in the past. All the material particles present in an arbitrary neighborhood of x must be considered. To this it was added the explicit dependence upon the wall configuration, as a functional of s and of t. Not only, the present position of the wall is of importance, but also positions in the recent past. Notice additionally, that H depends on the entire surface as its defining variable ranges over the set S_{t}. Under a change of frame the following applies as an objectivity postulate:
(4) 
In the above equation, the composition sign indicates the action of changes of frame upon scalars, vectors, or tensors, as appropriate for the specific variable represented by H.
As the wall configuration and the fluid motion are simultaneously changed in consequence of the change of frame the response functional assumes objective values. It is therefore not true that H is an isotropic function of the motion, or of kinematic variables deduced from the motion such as velocity gradient, rate of strain, spin, etc. As the surface is altered in changes of frame, the functional dependence on the motion changes, requiring a representation for a functional obeying (4). It is not difficult to demonstrate that this representation can be expressed as:
(5) 
The relative motion with respect to the linear displacement of the moving surface, for both variables appearing in the right hand side of eq. (5) are the appropriate objective variables for the closure equation. Objectivity is easily verified by observation of eqs. (1) and (2). Conversely, if Q = 1, and r = c are introduced in eq. (4) the representation given is established. This last equation represents a functional with respect to the variables £ t, s Î S_{tt}, and z ÎN(x), a neighborhood of x.
If the flow system present surfaces moving with different velocities, then it is necessary to generalize eq. (5) to:
(6) 
in which surface number one is chosen as the reference. This result is too general and will be transformed into simpler models.
RATE TYPE EQUATIONS
One simplified form of eq. (5) results from the assumption that the functional dependence reduces to the dependence on the time derivative of the two variables, and on their respective gradients.
(7) 
Upon the use of the previously given definitions, and of restrictions imposed by objectivity one arrives at the following closure equation.
(8) 
where
(9) 
is the relative velocity,
D is the rate of strain,
W is the fluid intrinsic spin,
W_{s} is the surface intrinsic spin.
For the case of multiple surfaces with different velocities, this equation is written as:
where 
(10) 
Dependence on the velocity field is perfectly compatible with frame indifference, as long as relative velocities are considered, and the surface provides a special reference. It will be shown the usefulness of allowing the relative velocity to be included on the list of state variables. The rate of strain is the main variable of all models, while the dependence on the fluid spin is now common practice. The surface spin is justified by the fact that it is the determinant variable for the velocity of rotation of the surface. It might be argued that if the frame of reference were made solidary with the rotating surface then the surface spin would be absent. However, restriction to the extended Galilean group forbids this change of frame.
The gradient of the surface is the new variable. It contains information sufficient to allow the calculation important surface characteristics, such as the unit normal and the first fundamental form.
(11) 
In this expression, g is a symmetric tensor over the twodimensional vector space in the tangent plane at a given position on the surface. It defines the positivedefinite quadratic form for the infinitesimal arc length. It possesses two positive eigenvalues, which correspond to two orthonormal eigenvectors in the tangent plane. The cross product between these gives the unit normal. Let A be the symmetric tensor in threedimensional space with a unit eigenvalue associated to the unit normal and additional eigenvectors and eigenvalues coinciding with those of g. This tensor is an incomplete local representation of the surface. The second fundamental form, which contains information about the surface curvatures, will not be used in the present work.
Equation (8) becomes:
(12) 
This expression for the closure of turbulent fluxes contains a set of kinematic variables, and a single tensor function describing the surface. Its dependence on the surface position variable will not be made explicit. Objectivity still applies,
(13) 
In order to be able to write valid representations for the function in eq. (12) it is necessary to consider the symmetries of the surfaces involved. The symmetry group of a surface is defined as the set:
(14) 
This will be referred to as the symmetry group of the surface.
Consider as the first case the flow about a sphere centered on c. The symmetry group for this surface is the complete proper orthogonal group. In this case, and only in this case the closure equation is determined by an isotropic function of the state variables. Furthermore, the surface can be replaced by its diameter as its single significant parameter.
(15) 
One can comment that the functional dependence on the surface variable x is absent from eq. (15). Representation theorems (the reader is referred to the work of Smith (1971) exist for this case, and their expressions are widely used. Inclusion of velocity gives added flexibility and, in fact, it may be of importance in adjusting the equation near the wall. The spherical symmetry gives rise to the simplest expression for the turbulent properties. Algebraic expressions for the Reynolds stress as a function of the shear rate, in some cases with the inclusion of the spin, are the only ones presently employed. It is now clear that these expressions are rigorously valid only for spherical symmetry. All the other surfaces possess smaller symmetry groups, giving rise to more complex representations. As a rule it is observed that the smaller the group, the less restrictions it imposes on the closure relation, giving rise to more complex expressions. This fact becomes evident when one considers the case of the flow near a plane infinite wall, or between parallel plates. In this case the symmetry group,G(S), contains all rotations about the normal to the planes. Dependence on the surface reduces to dependence upon the unit normal and the closure equation becomes:
(16) 
H is an objective function on the variables shown. Similar results are applicable to the case of flows inside infinite or semiinfinite pipes, with cylindrical walls or even in the annular region between two concentric pipes. For such cases, the symmetry group contains all rotations about the unit vector in the direction of the cylinder axis, e.
(17) 
In both these cases, the closure equation contains one additional vector, and the representation theorems for isotropic functions will present additional terms. It must be emphasized that the vectors characterizing these surfaces are of different orientations with respect to the flow fields, generating different expressions for the stresses in the internal viscometric flows. The differences disappear for the simpler viscometric flows, but may be of importance in the entrance region.
The functional dependence on the surface position disappears in the forms applicable to flows near these simple surfaces. A possible form for the functional relationship is that of a local average value integrated over the whole surface
(18) 
where W_{G }is the kernel for the averaging operator, a scalar function decreasing with the distance between the variable point in space x, and points on the surface,x . It should reduce to the Dirac d functional if x approaches the surface. In this case, eq. (18) is written as:
(19) 
This equation represents the general form of the model proposed, incorporating kinematic variables pertinent to the fluid motion, and the surface characteristics; the unit normal, the arc length. The surface characteristics are given by local averages, integrated over the whole surface.
Surfaces such as half planes, or nonsmooth surfaces as steps, square cavities, and many others, remain to be considered. These present edges, discontinuities, or angles, in general sets of points where the normal or the two fundamental forms are not defined. The set of irregular points has zero area, and do not contribute to the integral (18), and even in these cases the average value of the surface parameters can be calculated. An example is provided by the "backward facing step" of height h for which the unit normal varies as:
Consequently the average value of the unit normal taken at the position x, is:  
(20) 
As the weighing function decreases with distance, the local average of the unit normal is closer to e^{1} or to e^{2} depending on the position x.
TURBULENCE STRESS MODELS AND FINAL REMARKS
Discussion will be restricted to models for the turbulent stress, , but it must be kept in mind that the same comments apply to other turbulent fluxes, and to all the different correlation appearing in the different closure methods. In this case, equation (12) becomes:
(21) 
All the algebraic turbulence models for the Reynolds stress and other turbulent fluxes are special cases of eq. (21), where the surface characteristic parameters and dependence on the velocity field are omitted. In order to simplify matters, attention will be focused on stationary surfaces. In complex flows, particularly about rotating surfaces, the spin must be added.
(22) 
Comparison will be based on SSG model proposed by Gatski and Speziale [8]. This is an improved form of ke model based on a modeled closure of the complete equation for the transport of the Reynolds stress. The model obeys the following equations.
(23) 
where cm is a constant, approximately equal to 0.09, k is the kinetic energy of the turbulent fluctuations, e the dissipation, and the coefficients a may depend on the scalar invariants of D and W. k , and e satisfy modeled transport equations. A possible expression derived from equation (22), in view of (23) is:

(24) 
A single term is added, in the form of the tensor product of the relative velocity. This term allows a difference in the normal stresses in viscometric turbulent flows, which is not predicted by equation (23) or any other form of ke model. The coefficients of this new form are functions of the combined invariants of A, v, D, and W. The surface characteristics will manifest only through proposed values for these coefficients, applicable to each class of different surfaces.
A more complex form of equation (22), compatible with the SSG model, and with an explicit dependence on surface parameters is:
(25) 
For the special cases of spherical symmetry a_{AD} must be zero, and equation (24) applies. For all other surfaces a_{AD} differ from zero. For the special cases of flows about planes and cylinders equations (16) and (17) must be taken in consideration, and one determines:
(26) 
(27) 
Viscometric flows for which there exists an orthogonal basis such that the velocity field is v = v(x_{2})e_{1}and its gradient possesses only one nonzero component L_{12} = g provide important test cases. Eqs. (26), and (27) generate the same component expressions for the turbulent stress.
REFERENCES
Gatski,T. B. and Speziale, C. G. On explicit algebraic stress models for complex turbulent flows. J. Fluid Mech. 254, 5978 (1993). [ Links ]
Müller, I. Frame dependence of stress and heat flux. Arch Rat Mech Anal 45, 185194 (1972). [ Links ]
Murdoch, A. I. Material frameindifference, intrinsic spin, and certain constitutive equations motivated by the kinetic theory of gases. Arch Rat Mech Anal 83, 185194 (1983). [ Links ]
Spalart, P. R. and Speziale, C. G. A note on constraints in turbulence modelling. J. Fluid Mech 391, 373376 (1999). [ Links ]
Speziale, C. G. A Review of material frameindifference in mechanics. Appl Mech Rev 51, 489504 (1998). [ Links ]
Smith, G. F. On isotropic functions of symmetric tensors, skewsymmetric tensors, and vectors. Int J. Engng Sci 9, 899916 (1971). [ Links ]
Truesdell, C. and Noll, W. Nonlinear field theories of mechanics. Handbuch der Physik III/3, Springer Verlag, New York. (1965). [ Links ]
Wang, L. Frameindifferent and positivedefinite Reynolds stressstrain relation. J. Fluid Mech. 352, 341358 (1997). [ Links ]