Anisotropy Properties of Turbulence

Transcription

1 Proceedings of the 3th WSEAS International Conference on APPLIED MATHEMATICS (MATH'8) Anisotropy Properties of Turbulence SANKHA BANERJEE Massachusetts Institute of Technology Department of Mechanical Engineering U.S.A OEZGUER ERTUNC & FRANZ DURST University of Erlangen-Nueremberg Institute of Chemical Engineering Germany Abstract: In the literature, anisotropy-invariant maps are being proposed to represent a domain within which all realizable Reynolds stress invariants must lie. It is shown that the representation proposed by Lumley and Newman has disadvantages owing to the fact that the anisotropy invariants (II, III) are nonlinear functions of stresses. In the current work, it is proposed to use an equivalent linear representation of the anisotropy invariants in terms of eigenvalues. A novel barycentric map, based on the convex combination of scalar metrics dependent on eigenvalues, is proposed to provide a non-distorted visual representation of anisotropy in turbulent quantities. This barycentric map provides the possibility of viewing the Reynolds stress and any anisotropic stress tensor. Additionally the barycentric map provides the possibility of quantifying the weighting for any point inside it in terms of the limiting states (one component, two component, three component). The mathematical basis for the barycentric map is derived using the spectral decomposition theorem for second-order tensors. In this way, an analytical proof is provided that All turbulence lies along the boundaries or inside the barycentric map. It is proved that the barycentric map and the anisotropy-invariant maps are one to one uniquely interdependent and as a result satisfy the requirement of realizability. Key Words: Nonlinearity, Barycentric map, Quantification of anisotropy. Introduction The momentum transport at a point in a complex three-dimensional turbulent ows is not uniform in all directions, since, this process is characterized by a tensor, namely Reynolds stress tensor. This tensor u i u j can be identified as normal (when i = j) and tangential (when i j) turbulent stresses. It is very difficult to gain complete information contained in the Reynolds stress tensor using only one scalar, so, it is necessary, to extract information about, the different aspects of the tensor using different scalars. The Reynolds stress tensor carry information about the componentality of the turbulence (i.e the relative strengths of different velocity uctuation components). From DNS studies of turbulent ows, it can be concluded that each large scale structure tends to organize spatially the uctuating motion in its vicinity, and in so doing, it tries to eliminate the gradients of the uctuation fields in certain directions (the directions in which the spatial structure is significant), and it enhances the gradients of uctuations in other directions. Thus associated with each eddy in a turbulent momentum transport are local axes of dependence and independence. In the undeformed state of isotropic turbulence the various axes of dependence and independence due to individual eddies are oriented randomly. The velocity uctuation field which can be thought to be ensemble of all the eddies has gradients in all three directions. In turbulent ows, the mean ow gradient creates structural anisotropy, since it stretches and aligns the energy-containing turbulent eddies. This in turn aligns the axes of dependence and independence of individual eddies and creates directions in which, in a statistical sense, the gradients of energycontaining uctuations are weak. Thus by acting on the energy-containing structure mean ow gradient creates axes of independence, which are reected in auto-correlations of gradients of the uctuation fields. Thus there are regions in a turbulent ow, where there is one such direction of independence, and the turbulence becomes two-dimensional. However the direction of independence, says nothing about the intensity of the uctuations in that direction relative to the total intensity. Thus the anisotropy in the dimensionality of the turbulence is in general distinct from the anisotropy of its componentality. So characterization of the more general state of the turbulence requires information about the dimensionality of the turbulence (i.e the relative uniformity of structures in different directions) as well as the componentality of the turbulence (i.e the relative strengths of different velocity uctuation com- ISSN: ISBN:

2 Proceedings of the 3th WSEAS International Conference on APPLIED MATHEMATICS (MATH'8) ponents). Turbulence models carrying only componentality information (e.g. standard Reynolds stress models carrying only Reynolds stress tensor) cannot possibly satisfy conditions associated with the dimensionality of the turbulence or even reect the differences in dynamic behaviour associated with structures of different dimensionality (i.e nearly isotropic turbulence vs. turbulence with strongly organized twodimensional structures as observed in stratified ows). This was the central message put forward from the work of Kassinos & Reynolds (994)[]. In the current work, the additional tensors introduced by Kassinos & Reynolds (994)[] will be plotted in the barycentric map and in the metrics plot introduced by Banerjee et al. (7)[]. Kinematic relationships will be derived between the additional tensors and the anisotropic Reynolds stress tensor to see how their information can add to the existing knowledge of turbulence modelling. Our analyis will be limited to the domain of irrotational homogeneous ows, so the inhomogenity tensor C ij and the stropholysis tensor Q ijk are not taken into account. Kassinos & Reynolds (994)[] showed that three kinds of information are important in determining the evolution of the turbulent stresses in homogeneous turbulence;. Componentality information embodied in u i u j.. Dimensionality information available through and f ij (definitions given below). 3. Information about the breaking of reectional symmetry by mean or frame rotation carried by the stropholysis tensor Q ijk. The additional tensors and f ij are defined in the following from Kassinos & Reynolds (994)[]. Dimensionality tensor The structure dimensionality tensor D ij, its associated normalized tensor dij and dimensionality anisotropy tensor are defined as D ij = ψ n,i ψ n,j = D ij D kk = δ ij 3 () where the turbulence stream function vector ψ i, is defined by u = ǫ its ψ s,t ψ i,i = ψ i,nn = ω i () where ω i denotes the turbulent vorticity vector. The D ij tensor reveals the level of two-dimensionality of the turbulence. For example, when the turbulence is independent of x 3 direction then d 33 = because none of the stream-function components varies in that direction. Another definition of D ij can be arrived at, by considering the inviscid rapid distortion theory of homogeneous turbulence, for which the evolution equations for the Reynolds stresses, R ij = u i u j, are (see for example Reynolds & Kassinos (994)[]) dr ij dt = G ik R kj G jk R ki + Π r ij (3) where G ij = U i,j is the mean velocity gradient tensor and Π r ij is the rapid pressure strain-rate term, which results from the familiar splitting of the pressure uctuations into slow and rapid parts (see equation?? for detailed understanding) where κk κ l M ijkl = κ E ij(κ)d 3 κ (4) where E ij (κ) = û i (κ)û j (κ) is the velocity spectrum tensor, κ is the wavenumber vector, the hats denote Fourier coefficients and the denotes a complex conjugate. M ijkl is a fourth-rank tensor that must be modelled in terms of the tensor variables in the one-point model. The two non-zero contractions of M ijkl are R ij = M ijkk = E ij (κ)d 3 κ κk κ l D ij = M iikl = κ E ij(κ)d 3 κ (5) Circulicity tensor The structure circulicity tensor F ij, and its associated normalized tensor f ij and the circulicity anisotropy tensor f ij are defined by F ij = ψ i,n ψ j,n f ij = F ij F kk f ij = f ij δ ij 3 (6) The F ij tensor describes the large-scale structure of the vorticity field. When, most of the large-scale circulation is about the x 3 axis then the large-scale circulation is concentrated about the x 3 axis, i.e f 33. It has to be noted from the above definitions, that the trace of F ij and D ij are equal. The tensors u i u j, D ij and F ij are all positive semi-definite, and have only two independent anisotropy invariants. When considered together, D ij and F ij give a fairly detailed description of the turbulence structure. The secondrank dimensionality and circulicity tensors represent one-point correlations that carry non-local information about the structure of a turbulent ow. This is due to the fact, ψ i satisfies the poisson equation. From equations () and (6) for homogeneous turbulence it can be derived that u k u k = D kk = F kk = q (7) ISSN: ISBN:

3 Proceedings of the 3th WSEAS International Conference on APPLIED MATHEMATICS (MATH'8) and a fundamental constitutive equation in homogeneous ow can be derived of the form u i u j + D ij + F ij = q δ ij (8) From equation (8) it can be found that only two tensors are linearly independent in the context of homogeneous turbulence. By carrying only u i u j as the model variable, Reynolds stress models effectively lump together information found in the dimensionality D ij and circulicity F ij into u i u j. It is because of this fact, the Reynolds stress models are not sensitive to the differences in the dynamic behaviour associated with the different dimensionality in homogeneous turbulence.. Properties of the Reynolds Stress and its Anisotropy Tensor The symmetry property of the Reynolds stress tensor u i u j ensures diagonalizability of its matrix. In the diagonal form, all tensor entities of u i u j are the matrix s eigenvalues, which are non-negative by definition. The individual components of the Reynolds stress tensor, have to satisfy the physical realizability constraints (9) noted by Schumann[3] u µ u µ, u µ u µ + u ν u ν u µ u ν, det (u i u j ), µ,ν = {,,3}. (9) From equation (9), it can be concluded that the Reynolds stress tensor is required to be a positive semi-definite matrix. From equation (??) and the condition of non-negativeness for the main diagonal elements in equation (9) the following relationship holds for the corresponding anisotropy tensor b ij b µµ = u µu µ k, µ = {,,3}. () 3 It can be seen that the diagonal component of a µµ of the anisotropy tensor takes its minimal value if u µ u µ =. The maximal value corresponds to the situation when u µ u µ = k. For the off-diagonal component a µν of the anisotropy tensor, one can write a similar expression: b µν = u µu ν, µ ν, µ,ν = {,,3}. () k Because of the positive semi-definiteness of u i u j, the off-diagonal element a µν reaches its minimal value in the case when u µ u ν = k and its maximal value if u µ u ν = k. Applying these conditions to equations () and (), intervals between which the anisotropy tensor components belong can be calculated: /3 b µµ /3, / b µν /, µ ν, µ,ν = {,,3}. (). Canonical Form of the Anisotropy Tensor The anisotropy tensor of Reynolds stress is also symmetric i.e, b ij =b ji. Its eigenvalues are real and the eigenvectors associated with each eigenvalue can be chosen to be mutually orthogonal. Hence the following orthogonality condition holds for the eigenvectors: x k i x l i = δ kl (3) Furthermore, if a tensor X ij consists of the eigenvectors of b ij such that the orthogonality condition (3) holds, then the tensor b ij can be diagonalized by similarity transformation: where X ik b kl X lj = ij (4) ij = λ i δ ij (5) is a diagonal tensor with the eigenvalues of b ij on its diagonal. The coordinate system made of the eigenvectors is termed the principal axes in which b ij is diagonal. The relationship between the Reynolds stress tensor and the anisotropic stress tensor is given by the relation (??). Let λ ri be an eigenvalue of u i u j and λ i be the corresponding eigenvalue of b ij. Let the corresponding eigenvector be x i. Then b ij x i λ i x i = u iu j k x i δ ij 3 x i = λ ri k x i δ ij 3 x i ri,i = {,,3} (6) where λ i = λ ri k 3, with eigenvectors of u iu j and b ij being x i. It is important to note that the eigenvectors of u i u j and b ij are identical. This information makes the trajectory of the normalized Reynolds stress and the anisotropic Reynolds stress identical inside the barycentric map for any kind of ow. In the current work, the following notation for the eigenvalues of the anisotropy tensor after diagonalization (i.e in principal coordinate system (pcs)) are applied: λ = max µ (b µµ pcs ), λ = max ν µ (b νν pcs ), µ,ν = {,,3}. (7) The scalar functions λ and λ are the two independent anisotropy eigenvalues that can be used for the characterization of the anisotropy stress tensor. From the fact that the anisotropy tensor has a zero trace, one can represent its smallest eigenvalue by means of the above introduced notation: λ 3 = λ λ = min χ =µ,ν (a χχ pcs ). (8) ISSN: ISBN:

4 η A Proceedings of the 3th WSEAS International Conference on APPLIED MATHEMATICS (MATH'8) From the notation (7), the following inequality holds: λ λ λ 3. (9) Introducing the non-increasing order on diagonal elements of the anisotropy tensor b ij, it can be considered by the reordered matrix ˆb ij. It can be seen that for any physically realizable Reynolds stress tensor, it is possible to construct exactly one corresponding anisotropy tensor in its canonical form () that can be written in terms of notation (7) as ˆbij = λ λ λ 3.3 Anisotropy-Invariant Maps. () The anisotropy of turbulence can be characterized by the following techniques. The first representation, proposed by Lumley and Newman[4], given in figure (??, (left)), can be constructed from the following nonlinear relationships based on the two invariants of the anisotropy tensor (), denoted II III: I = b ii II = b ij b ji III = b ij b in b jn () II 3 ( ) 4 /3 3 III, II + III. () 9 The second representation suggested by Lumley[5], denoted η ζ is an alternative representation of II III, denoted η ζ: ζ 3 = III/ η = II/ (3) This map is depicted in the figure (), (right). The third representation was originally suggested by Lumley[6], based on the eigenvalues of the anisotropy tensor, denoted λ,λ, depicted in the figure, (left). This representation and its relationships, were used in the modelling of turbulence by Terentiev. The relationships of the different boundaries of this map in terms of λ,λ : λ (3 λ λ )/, λ /3 λ (4) It is easy to see that there exists a direct functional relationship between the anisotropy invariants and its corresponding eigenvalues [i.e. II = w(λ,λ ) and III = q(λ,λ )]. For the transformation from the eigenvalue representation (λ,λ ) to the principal components (III, II), one can directly use the definition for the anisotropy invariants expressed by equation () to obtain B II = (λ + λ λ + λ ), III = 3λ λ (λ + λ ). (5) The eigenvectors V, V and V 3 are column vectors which describe the orientation of the momentum One component limit (A=/3, B= /3) Two component limit (A=/3 B) Axisymmetric expansion limit (A=(3 B B)/) Two component plane strain limit (A=/3, B=) Plane strain limit (B=) Two component axisymmetric limit (A=/6, B=/6) Axisymmetric contraction limit (A=(3 B B)/) Isotropic limit (A=, B=) C, axi axi, ζ < C axi, ζ > iso ζ Figure : Anisotropy-invariant map based on two independent eigenvalues of the anisotropy tensor (A, B) (left) and anisotropy invariant map based on (ζ, η) (right). The eigenvalues (A,B) correspond to λ,λ..4 Anisotropy invariant measures The different aspects of the Reynolds stress tensor which can be considered to be informative are listed as follows:. The overall magnitude of momentum transport.. The degree, with which the rate of momentum transport is dissimilar, in different orthogonal directions. This is referred to as the degree of anisotropy. Information about the relative magnitude of the eigenvalues of the Reynolds stress tensor, i.e the largest, intermediate, and smallest. This is referred to as the eigenvalue order 3. The degree, that the rate of momentum transport in two of the three orthogonal directions differs from the rate of momentum transport in the third direction, i.e, how symmetrically the momentum transport is distributed in the orthogonal directions. This can be referred to as the skewness of the momentum transport. The anisotropy and the skewness measures of the Reynolds stresses are non-dimensional, i.e they are independent of scale. The diagonal elements of the Reynolds stress tensor, describe the rate of momentum transport in the different directions and the offdiagonal elements describe the degree of correlation of the momentum transport along the various axes. As, a first step, in extracting scalar information from the Reynolds stress tensor, it is necessary to divide it into two parts, each having complementary information. Diagonalization of the Reynolds stress tensor seperates it, into its component eigenvalues and eigenvectors. u i u j = [ V V ] V 3 λ λ λ 3 C V V V 3 ISSN: ISBN:

5 Proceedings of the 3th WSEAS International Conference on APPLIED MATHEMATICS (MATH'8) Set e = j j e j for j =;, then: Where F ( ^X ) = F ( ^X ) F ( ^X ) = F ( ^X ) _e = F ( ^X ) S CT C e o + nf ( ^X )X F (X )X + n G (U ; ^X ; ^X ) G (U ;X ;X ) () _e = F ( ^X ) S CT C e o + nf ( ^X )X F (X )X + n G (U ; ^X ; ^X ) G (U ;X ;X ) () Lemma If the hypothesis are satisfied, then the system ()-(3) is an exponential observer of the system (4)-(5) for and are positives. 3. Proof of the lemma To prove convergence, let us consider the following equation of LYAPUNOV: V = V + V () where V = e T S e and V = e T S e. By calculating the derivative of V along the trajectories of e and e, we obtains: _V = e T S _ e +e T ΛT S _ Λ e +e T S e _ +e T ΛT S Λ _ e = e T S F ( ^X )e e T CT C e +e T S +e T S nf ( ^X )X F (X )X o n o G (U ; ^X ; ^X ) G (U ;X ;X ) +e T ΛT S Λ _ e + e T S F ( ^X )e e T CT C e +e T S +e T S nf ( ^X )X F (X )X o n o G (U ; ^X ; ^X ) G (U ;X ;X ) +e T ΛT S _ Λ e (3) By taking account of the hypothesis, and by introducing the norms, then: C _V» V e + S e n o F ( ^X )X F (X )X o o + S e n G (U ; ^X ; ^X ) G (U ; X ;X )gk +e T ΛT S Λ _ C e V e S e n o F ( ^X )X F (X )X S e n G (U ; ^X ; ^X ) + + G (U ;X ;X )gk +e T ΛT S Λ _ e (4) n F ( ^X )X n» V +» max(s ) ke k n F (X )X gk + G (U ; ^X ; ^X ) G (U ;X ;X )gkg +» max (S ) ke k n F ( ^X )X n V +» max(s ) ke k n F (X )X gk + G (U ; ^X ; ^X ) G (U ;X ;X )gkg +» max (S ) ke k» V + c V V + c V (5) where Λ = sup Λ _ and =sup Λ _ Λ min (S ) and max (S ) ( min (S ) and max (S )) are the eigenvalues minimal and maximum of S (S ) respectively. Λ» =max(;fi ) and» = max ;fi ;fi. By rewriting the preceding expression of V () according to V and V, one obtains: Where _V» ( c )V ( c )V (6) c =ff(s )»» (ο + ο +» ) ;» = min (;ff ) ; c =ff(s )» s» max (S ) ff(s )= min (S ) If we take: 8> < >: Finally, while taking: It follows that: (ο 3 + ο 4 +» ) ;» = min ;ff ;ff ; ß = c > and ß = c > ; ff(s )= = min (ß ;ß ) (7) _V» (V + V )» V (8) This finishes the proof of convergence of the observer. Thus V is a LYAPUNOV FUNCTION. s max (S ) min (S ) : ISSN: ISBN:

6 Proceedings of the 3th WSEAS International Conference on APPLIED MATHEMATICS (MATH'8) the turbulent momentum transport in orthogonal directions. But, they are not invariant as, λ λ λ λ (7) λ 3 λ 3 Cartesian coordinate with ordered eigenvalues. In this coordinate system, the position of a point A is specified by the eigenvalues of the Reynolds stress tensor ordered by magnitude. A ordered (λ,λ,λ 3 ) = λ max λ int λ min (8) where λ max,λ int, and λ min denote the maximum, intermediate and minimum values of the eigenvalues. This is not a true coordinate system, since it maps six points from eigenvalue space onto one point in the ordered eigenvalue space. The six points, which are mapped onto ordered eigenvalue space, are points defined by the six possible permutations of the eigenvalue order. However, A ordered transformation is invariant. Rotated cartesian coordinate system. The axes of the original (not ordered) Cartesian coordinate system can be rotated by taking linear combinations of eigenvalues. The rotation matrix must have orthogonal basis vectors in order for the new coordinate system to have orthogonal axes. If the rotation matrix is taken to be orthonormal (i.e the orthogonal basis vectors are of unit length) then there is no resultant dilation or contraction of the eigenvalue space. When an orthonormal transform is used, the resulting measures maintain the units of the original cartesian system. A rotated (λ,λ,λ 3 ) = θ θ θ 3 θ θ θ 3 θ 3 θ 3 θ 33 λ max λ int λ min A particularly useful orthonormal rotation matrix is one which places the new z axis along the eigenvalue triple identity line, λ = λ = λ 3, with the positive z axis in the direction of positive eigenvalues. When the turbulent momentum transport in the Reynolds stress tensor is isotropic, i.e the same in all directions, the eigenvalues of the Reynolds stress tensor lie along this triple identity line. A r = λ λ λ 3 = x y z (3) where x, y, z >.. The eigenvalue triple identity line is an obvious line of symmetry in the eigenvalue (9) space, since any measure applied to any point along the line of triple identity will automatically be invariant. The subscript r in A r denotes the particular rotation of the axes where the z axis is along the triple identity line. In this system the x-axis is arbitrarily chosen to be along λ = λ 3 plane with the positive x axis in the direction where λ < λ, (see figure ()). The positive y axis is in the plane λ = (λ + λ 3 )/ with positive y direction where λ < λ 3. As shown in Eqs. (3), z is an invariant, whereas x and y are not invariant measures. The parameter z has the properties of trace of a matrix u i u j and λ bar which is the mean of the three eigenvalues. z = (λ + λ + λ 3 ) 3 = λ bar 3 (3) lambda=lamda3 p=3 p=4 p= p=5 y axis The six points in the eigenvalue space correspondp= ordered space lambda=lambda p=6 lambda=lambda3 Figure : The plane of the six permutations of the eigenvalue order. ing to the six permutations of the eigenvalues of the Reynolds stress tensor, when defined in terms of the rotated cartesian coordinate system, all have the same z value. These points defined by the six permutations of the eigenvalues display an interesting symmetry in the x y plane (orthogonal to the z axis). This six points all lie at a fixed distance r from the orign in the x y plane (see figure(3)). This radial symmetry of the eigenvalue permutations in the rotated cartesian coordinate system can be utilized in circular cylindrical coordinate system described below. The six points defined by the six permutations of the eigenvalues are considered to be three sets of two points; each set is symmetrically placed about one of the three lines defined by the intersection of the x,y plane and the planes: λ = λ, λ = λ 3 and λ 3 = λ in figure (). Circular cylindrical coordinate system. This system is a modification of the A r system, Eq. [3]. The x axis ISSN: ISBN:

7 Proceedings of the 3th WSEAS International Conference on APPLIED MATHEMATICS (MATH'8) deviation of the eigenvalues. lambda=lambda3 lambda=lamda sd(λ,λ,λ 3 ) = (λ λ bar ) +(λ λ bar ) +(λ 3 λ bar ) (36) = r (37) r lambda=lambda3 x axis Figure 3: The x y plane of the rotated Cartesian coordinate system showing projection of points defined by the six permutations of the eigenvalue order. These points all lie in at a distance r from the orign. z axis is kept the same in the A r system. The other two coordinates, r and φ of the A cyl are the x and y of A r written in polar coordinate form. λ bar denotes the mean of the eigenvalues Eq. [3], r, and φ < π. P cyl (λ,λ,λ 3 ) = r φ z (3) The value of the polar angle measure, φ must be defined to account for, which quadrant of the x-y plane, the eigenvalue point lies in figure (). Case I, Quadrant I: x > and y > then φ(λ,λ,λ 3 ) = arctan [ 3(λ λ 3 ) λ λ λ 3 Case II, Quadrant II and III: x < φ(λ,λ,λ 3 ) = π + arctan ] [ 3(λ λ 3 ) λ λ λ 3 Case IV, Quadrant IV: x > and y < φ(λ,λ,λ 3 ) = π + arctan [ 3(λ λ 3 ) λ λ λ 3 ] ] (33) (34) (35) As in the A r system, z is invariant. Equation [??] demonstrates that the new parameter r is also invariant. r can be shown to be proportional to the standard where sd(λ,λ,λ 3 ) is the standard deviation of the eigenvalues. The polar angle measure φ, is unitless but not invariant. The zero point of φ is chosen to be x axis, where λ = λ 3, in the direction that λ < λ. φ ranges from to π. It is important, to note from the above measure that a scalar is not invariant by definition. Another invariant measure, which represents the variation of the momentum transport in terms of the eigenvalues of the Reynolds stress tensor is cv(λ,λ,λ 3 ) = = 3r z (38) (λ λ bar ) +(λ λ bar ) +(λ 3 λ bar ) λbar (39) where cv(λ,λ,λ 3 ) is unitless. The cv(λ,λ,λ 3 ) measures are not orthogonal to z. Thus the values of cv(λ,λ,λ 3 ) and z are neither independent nor complementary. Spherical coordinate system. This coordinate system is composed of one radial measure, ρ, and two unitless angle measures, θ and φ. The polar angle measure φ of the spherical coordinate system is identical to that of circular cylindrical coordinate system: A s (λ,λ,λ 3 ) = ρ(λ,λ,λ 3 ) = [ θ(λ,λ,λ 3 ) = arccos φ(λ,λ,λ 3 ) = γ + arctan θ φ ρ (4) λ + λ + λ 3 (4) (λ +λ +λ 3 ) 3 λ +λ +λ 3 ] [ = arccos z ρ = arctan [ ] r z [ 3(λ λ 3 ) ] λ λ λ 3 ] (4) (43) (44) (45) where ρ >, and φ < π. The quadrant modification γ was detailed in the Eqs.[33-35]. Since, ρ is invariant and is the measure of the total magnitude of momentum transport. θ is unitless and invariant. It can be related to the coefficient of variation of the ISSN: ISBN:

8 Proceedings of the 3th WSEAS International Conference on APPLIED MATHEMATICS (MATH'8) eigenvalues and thus is a measure of the relative dispersion. φ is unitless but not an invariant. It is an indicator of the permutation of the order of the eigenvalues which is discussed below. The zero point of θ is chosen to coincide with the eigenvalue triple identity line (λ = λ = λ 3 ). The zero point of φ is again chosen to coincide with the plane λ = λ 3 in the direction that λ < λ. follows: p = λ < λ λ 3 p = λ λ < λ 3 p = 3 λ < λ 3 λ p = 4 λ 3 λ < λ p = 5 λ 3 < λ λ p = 6 λ λ 3 < λ (47) The skewness measure can be defined as λ =λ 3 p=3 p=4 p= λ =λ p= p=6 λ =λ 3 p = s = φ p = s = (π/3) φ p = 3 s = φ (π/3) p = 4 s = (4π/3) φ p = 5 s = φ (4π/3) p = 6 s = π φ (48) Figure 4: The x y plane. The lines indicate the intersection of the x y plane with the λ = λ, λ = λ 3, and λ 3 = λ planes, respectively. The region where p = corresponds to the projection of the ordered eigenvalue space onto the x y plane. The ˆbij distribution from data of direct numerical simulation of fully developed plane turbulent channel ow for = 8 by Kim et al.[7]. The region where p = corresponds to the projection of the ordered eigenvalue space onto the x y plane. p=5 The skewness parameter, s, can be equivalently be defined by ordered eigenvalues, [ ] 3(λint λ max ) s(λ,λ,λ 3 ) = arctan (49) (λ min λ int λ max ) The permutation p, skewness s or the angle φ can be used to check, in which barycentric map the ow trajectory lies. This is a very useful indicator of the eigenvalue order in turbulent stresses..5 General Classification of States of Turbulence The classification of turbulence used in the present study is based on the componentiality of turbulence, namely Reynolds stress tensor. The Reynolds stress s major, medium and minor axes are along the tensor s eigenvectors, with scalings along the axes being the The skewness measure, s and permutation indicator, p: The measure of the polar angle, φ, can be divided into two components. The first component is s, a eigenvalues. Let λ i 3 for i =,,3 be the major, medium and minor eigenvalues of the ˆb ij satisfy- skewness measure and the second component is p, an indicator of the order of the eigenvalues. The ing equation () and let x i be an eigenvector corroesponding indicator measure, p can be defined as follows for the figure (4) to λ i. From the spectral theorem, any second-order tensor can be written as p = φ < π/3 p = π/3 φ < π/3 p = 3 π/3 φ < π p = 4 π φ < 4π/3 p = 5 4π/3 φ < 5π/3 p = 6 5π/3 φ < π (46) ˆbij = λ x x T + λ x x T + λ 3x 3 x T 3 (5) The rank, i.e. number of non-zero eigenvalues, of the tensor ˆb ij reects the limiting states of turbulence present in the tensor ˆb ij. The limiting states of the second-order tensor can be characterized by the following states: where p is an indicator of the permutation of the eigenvalue order. An equivalent definition of p is as One-component limiting state of turbulence ˆ (b) c (one eigenvalue is non-zero in u i u j ), ISSN: ISBN:

9 Proceedings of the 3th WSEAS International Conference on APPLIED MATHEMATICS (MATH'8) Two-component limiting state of turbulence ˆ (b) c (two eigenvalues are non-zero in u i u j ), Three-component limiting state of turbulence ˆ (b) 3c (three eigenvalues are non-zero in u i u j ). In the current work, componentality reects the number of non-zero velocity uctuations u i, which matches the rank of the correlation matrix under consideration, e.g. for any anisotropic stress tensor. Special States of Turbulence of Anisotropy Tensor One-component limiting state This case corresponds to rank one tensor ˆb ij, where λ > > λ λ 3. This is the one-component (-c) state of turbulence with only one turbulence component. Turbulence in an area of this type is only along one direction, the direction of the eigenvector corresponding to the non-zero eigenvalue: ˆbij = λ x x T = λ ˆbc (5) The basis matrix for the limiting state is given by ˆbc = /3 /3 /3 Two-component limiting state (5) This case corresponds to rank two tensor â ij, where λ λ > > λ 3. This is the two-component (-c) isotropic state of turbulence, where one component of turbulent kinetic energy vanishes with the remaining two being equal. Turbulence in an area of this type is restricted to planes, the plane spanned by the two eigenvectors corresponding to non-zero eigenvalues: ˆbij = λ (x x T + x x T ) = λ ˆbc (53) The basis matrix for the limiting state is given by ˆbc = /6 /6 /3 (54) The introduced barycentric map can represent both componentiality and dimensionality of turbulence Three-component limiting state This case corresponds to rank three tensor ˆb ij, where λ =λ =λ 3. This is the three-component (3-C) isotropic state of turbulence. Turbulence in an area of this type is restricted to a sphere, the axes of the sphere are spanned by the eigenvectors corroesponding to non-zero eigenvalues: ˆbij = λ (x x T + x x T + x 3 x T 3 ) = λ ˆb3c (55) The basis matrix for this limiting state is given by ˆb3c = (56) ˆbij = C a cˆb c + C a cˆb c + C a 3cˆb 3c (57) C a c,ca c,ca 3c (58) where {Cc a,ca c,ca 3c } are the coordinates of the anisotropy } tensor â ij in the tensor basis {ˆbc,ˆb c,ˆb 3c. As described above, the relationships between the eigenvalues of the anisotropy tensor can be used for the classification of the ˆb ij by using the coordinates of the tensor in the new basis ˆbc, ˆb c and ˆb 3c to measure how close ˆb ij is from the generic cases of one component, two component and three component limiting states. To obtain a measure of anisotropy from the anisotropy tensor ˆb ij, metrics of three different kinds of anisotropy are provided. The scalar metrics are chosen in a manner such that C a c + Ca c + Ca 3c = (59) where Cc a is a linear measure of anisotropy and Ca c is the planar measure of anisotropy. The normalization is done in such a way that all metrics {Cc a,ca c,ca 3c } lie in the range [,]. It has to be noted that ˆb 3c is a null matrix which can make the equation (57) nonunique but the restriction on C3c a using (59) makes it unique. In general, any anisotropy tensor ˆb ij can be expressed as a convex combination (57) of the limiting states (one component, two component, three component) by setting C a c = λ λ C a c = (λ λ 3 ) C a 3c = 3λ 3 + (6) It means when you draw a line between two points in a given bounded domain, no part of the line falls out of the domain ISSN: ISBN:

10 a ij Proceedings of the 3th WSEAS International Conference on APPLIED MATHEMATICS (MATH'8) An anisotropy measure describing the deviation from isotropy is achieved as C a ani = Ca c + Ca c C a ani = Ca 3c C a ani = 3λ 3 (6) Taking this formulation into account a new map to characterize turbulence is proposed where the three limiting states of turbulence (one component, two component, three component) represent the three vertices of the map. In the current barycentric map, at each vertex one of the metrics {Cc a,ca c,ca 3c } has coefficient and then it decreases to zero for the line opposite to the vertex. A value of for any of the metrics corresponds to the respective limiting states and a value of means that the respective limiting state doesnot describe the current state. Any point inside the map has local barycentric coordinates {Cc a,ca c,ca 3c } which helps to know directly the weighting of the three limiting states of turbulence at that point and so quantification of the anisotropy is possible. Plotting of the Map The barycentric map can be used as a tool to visualize the nature of anisotropy in different types of turbulent ows. For plotting the barycentric map, we identify the basis matrices ˆb c,ˆb c,ˆb 3c of the equation (57) as the three vertices of the triangle. The corners of the map can be identified as three points (x c,y c ), (x c,y c ) and (x 3c,y 3c ) in the Euclidean space representing the three limiting states. To plot a point, we have to take a convex combination of the limiting states (6) using the equation (57): x new = Cc a x c + Cc a x c + C3c a x 3c (6) y new = Ccy a c + Ccy a c + C3cy a 3c (63) All ows can be plotted in the barycentric map figure 5 using the relations (6,63) to visualize the nature of anisotropy. An important point which needs to be noted in the construction of the barycentric map is that all the limiting states (one component, two component, three component) are joined by lines. The proof that these limiting states can be joined by lines is given in Terentiev (6)[8]. Another important feature of the barycentric map is that it can be constructed by taking three arbitrary basis points such as (x c,y c ), (x c,y c ) and (x 3c,y 3c ) so one can choose the basis of figure, (left) to obtain the eigenvalue map. The barycentric map is depicted in the form of an equilateral triangle as all the metrics (6) scale from [,] and an equilateral triangle doesnot introduce any visual bias of the limiting states (one component, two component, three component). comp Axi-symmetric contraction Plain strain Two component limit Axi-symmetric expansion comp Figure 5: Barycentric map based on scalar metrics which are functions of eigenvalues of the secondorder stress tensor describing turbulence. The isotropic point has a metric C3c a =, the twocomponent point has Cc a = and the one -component point has Cc a =..6 Computation of the Barycentric Coordinates The scalar metrics C c,c c and C 3c are computed from relations (6). Figure 6 shows the ˆb ij distribution for a fully developed plane turbulent channel ow at = 8. Figure 6 clearly illustrates that the mathematical requirement of axisymmetry ˆb = ˆb 33 is not satisfied, but when it is viewed in the AIM Fig. 7, (left) it gives a visual impression of axisymmetry y + Figure 6: The ˆb ij distribution from data of direct numerical simulation of fully developed plane turbulent channel ow for = 8 by Kim et al.[7]. The ˆb 33 stresses are % more than the ˆb stresses near the wall of the channel and meet the axisymmetry criterion only at the channel centreline. a a a 33 ISSN: ISBN:

11 II II Proceedings of the 3th WSEAS International Conference on APPLIED MATHEMATICS (MATH'8) x =h + x = III x + = h + x = comp comp Figure 7: Invariant characteristics of the anisotropy tensors for the Reynolds stress [ ] - u i u j on the anisotropy-invariant map (III, II) (left) and scalar metrics (C c,c c ) based on the eigenvalues of anisotropy tensor for the Reynolds stress on the barycentric map (right). Data were extracted from a direct numerical simulation of fully developed plane turbulent channel ow for = 8 by Kim et al.[7]. The barycentric map Fig. 7, (right) is much more consistent in preserving this physical information and it shows that the fully developed plane turbulent channel ow for = 8 is far from axisymmetry. In Fig. 7, (right) it can be observed that the barycentric map offers a visual advantage in depicting that the channel ow is far from isotropy at the channel centreline. The weighting of the limiting states for plane channel ow data of Kim et al.[7] were computed and it was found that the isotropic metric C 3c is.9 and not to indicate full isotropy of the ow at the channel centreline. Table gives a clear mathematical justification that the channel ows are not isotropic at the centreline as claimed by Kim et al.[9]. The nonlinearity in (II, III) makes the ow tend towards isotropy visually when viewed in the AIM (Fig. 7, left). Similar conclusions can be drawn about pipe ows when they are viewed in the barycentric map. From Fig. 8, (left), it is important to observe that when the point P, when projected to any one side of the barycentric map, say for instance, the two-component side, the distance from the two-component vertex to the line C c =.5 intercepts is the quantitative value of C c ; similarly the distance from the one-component vertex to the line C c =.5 intercepts is the value of C c. The value of C 3c can be obtained from the equation (59) and in the map it is the distance between the line joining the one and two-component states and the line where C 3c =.5 along the sides of the barycentric map. The manner in which the barycentric coordinates scale in the barycentric map is shown in Fig. (8), (right). The centre of the barycentric map corresponds to the point P where C c = 3, C c = 3, C 3c = 3 (64) The centre of the barycentric map when mapped back comp Points C c C c C 3c P x + x + = h...9 = Table : Weightings of the point P. C 3c =.5 C c =.5 P C c =.5 comp comp C3c Cc isotropic Figure 8: Calculation of the weighting of a point P from data extracted from a direct numerical simulation of fully developed plane turbulent channel ow for = 8 on the barycentric map (left) and the axis of the metrics C c, C c and C 3c on the barycentric map (right). The isotropic point corresponds to 3comp. on to the invariant map (AIM) gives a point very close to the axi-symmetric expansion curve which can be verified by inserting the barycentric coordinates (64) into (67). This fact can be visually noticed in the invariant map (AIM), Fig. (9), (left) and in the barycentric map Fig. (9), (right). This is a mathematical justification about the fact, why channel ows which are far away from the axi-symmetric limiting states in the barycentric map Fig. 7, (right) and are so near to the axi-symmetric limiting states in the AIM Fig. 7, (left)..7 Mapping Between the Barycentric and the Invariant Map In this section we show that there is a bijection between the barycentric map and the invariant map P III Cc Cc P Cc C3c comp comp Figure 9: The centrepoint P of the barycentric map (right) projected onto the invariant map (left). The point P lies closer to the axisymmetric states in the invariant map. comp ISSN: ISBN:

12 Proceedings of the 3th WSEAS International Conference on APPLIED MATHEMATICS (MATH'8) The domain of the invariant map (AIM) is bounded by the line II = 3 (4 3 III) 3 from below and by II = III + 9 from above. In other words, a point (III,II) lies inside the invariant map if and only if 3 (4 3 III) 3 II III + 9. (65) Let the barycentric coordinates C c,c c and C 3c of the barycentric map be given, and from the definitions (6) we obtain λ = C c + C c + C 3c 3 3, λ = C c + C 3c 3 3, λ 3 = C 3c 3 3. (66) Substituting C 3c = C c C c we deduce II = + ( 3 C c + ( c) 6 C + 3 C c + 6 C c ( 3 C c ) 3 C c = 3 C c + 3 C c C c + 6 C c, ( III = 3 C c + 3 ( c) 6 C + 3 C c + 6 C c ( + 3 C c ) 3 3 C c ) ) 3 = 9 C3 c + 6 C c C c C c C c 36 C3 c. This means that we can express II and III in terms of the barycentric coordinates C c, C c, and C 3c and this defines a mapping (III,II) = Φ(C c,c c ) from the barycentric map into the II III plane of the invariant map..8 Range of the Mapping In order to show that the range of Φ lies within the bounds of the invariant map (65), let C c, C c and C 3c be the barycentric coordinates of some point in the barycentric map, using the equation (67), we deduce and ( 3 II ) 3 ( 4 ) 3 III 4 = 7 C4 c Cc C3 c C3 c + 7 C c C4 c III + 9 II = 4 9 C3 c + 3 C c C c 6 C c C c 8 C3 c 3 C c 3 C c c C 6 C c + 9 = C 3c C c (C c + ) C c + ) 9 C 3c ( + C c + C c. Therefore, we have 3 (4 3 III) 3 II III+ 9 and thus the point (III,II) lies inside the domain of the invariant map (AIM)..9 Mapping from the Invariant Coordinates to the Barycentric Coordinates Let (III, II) be a point in the invariant map, fulfilling relation (65). We have to show that there are barycentric coordinates C c, C c, and C 3c fulfilling the relation (III,II) = Φ(C c,c c ). Consider the cubic function which defines the relation between (III,II) and (λ,λ ): f(x) = 3x 3 3 II x III. (67) The critical points of f are at x c = 6 II and x c = + 6 II with f(x c) = + 6 II3 III and f(x c ) = 6 II3 III. From relation (65), it follows that f(x c ) and f(x c ). Therefore, f must have exactly one root β with 6 II β + II (68) 6 and from equation (67) 3β 3 3 II β = III. (69) Choosing α = + II 3 4 β β, γ = (α + β). (7) then from the relation (68) it follows that II 3 4 β 3 II (7) 8 Hence α is real and β + 6 II α. In a similar way, it can be proved that γ II β. The estimate 3 γ can be derived from II III ISSN: ISBN:

13 II Proceedings of the 3th WSEAS International Conference on APPLIED MATHEMATICS (MATH'8). Uniqueness in the Mapping from the Barycentric map to the Invariant map Let (c c,c c,c 3c ) and ( c c, c c, c 3c ) be the barycentric coordinates of two points in the barycentric map, fulfilling relations (58) and (59) respectively, that are mapped to the same point (III,II) in the anisotropy map by Φ. That is, we have (III,II) = Φ(c c,c c,c 3c ) = Φ( c c, c c, c 3c ) (7) with Φ as defined by 67. In order to prove the injectivity of Φ, we have to show that (c c,c c,c 3c ) = ( c c, c c, c 3c ). Let α = 3 c c + 6 c c, α = 3 c c + 6 c c, β = 3 c c + 6 c c, β = 3 c c + 6 c c, γ = 3 c c 3 c c, γ = 3 c c 3 c c. (73) Note that we have α+β +γ =, α+ β + γ =, and by equation 58 α β γ, α β γ. Using this definition, we can write equation 7 as Note that II = α + β + ( α β) = α + β + ( α β), III = α 3 + β 3 + ( α β) 3 = α 3 + β 3 + ( α β) 3. (74) 6 II β = 6 c c c c (75) [ and thus β 6 II,+ 6 ]. II Similarly, β [ 6 II,+ 6 ]. II Combining equations 74 we obtain 3 β II + III = 3β3, 3 β II + III = 3 β 3. (76) In other words, β and β are both roots of f(x) = 3x 3 3 IIx III. We already checked above that comp A isotropic B comp A B III Figure : Linear interpolation between two points inside the barycentric map projected onto the invariant map. The isotropic point corresponds to 3comp. f has only one root in the range and therefore we get [ ] 6 II,+ 6 II β = β. (77) From equation 74, we obtain α = ± II 3 4 β β. (78) Insertion into γ = α β yields γ = II 3 4 β β. (79) Observing that α γ and using the same arguments on α and γ, we conclude that α = + II 3 4 β β = + II 3 4 β β = α (8) and finally γ = α β = α β = γ. Now we can deduce from equation 73 that c c c c (8) = α β = α β = c c = (β γ) = ( β γ) = c c c 3c = 3γ + = 3 γ + = c 3c (8) This completes the proof of the injectivity of Φ.. Important Features of the Barycentric Coordinates Figure describes an easy way of obtaining the interpolation between any two points on the AIM exactly. Taking a point having coordinates (II, III) in ISSN: ISBN:

14 II λ 3 Proceedings of the 3th WSEAS International Conference on APPLIED MATHEMATICS (MATH'8) the AIM we can find the eigenvalues corresonding to it by using equation (67). Once the eigenvalues are known, the barycentric metrics (6) can be computed. So if the linear interrelationships between unknown second-order tensors are desired, as in Jovanović et al.[] then the two points can be joined by a line in the linear domain of barycentric map and then converted back to invariants using equation (67). This provides a consistent and sound mathematical interpolation schemes between two points in the invariant map. Figure plots the barycentric metrics C c, C c and C 3c on the y-axis and y + on the x-axis from direct numerical simulation data of fully developed plane turbulent channel ow for = 8[7]. This D plot provides quantitative information about how ˆb ij behaves in terms of the limiting states (onecomponent, two-component and three-component) locally. Figure also provides the information that any point inside the barycentric map Fig. 5 bears the same area ratio (i.e take any point in the barycentric map or the eigenvalue map of Lumley and join that point with the limiting states of the map, the ratios of the areas of the three sub-triangles to the total area of the triangle is referred to as area ratio) as in the Lumley triangle Fig., (left). Figure provides the possibility of studying the dominance of the C c and C c in the near-wall region and the dominance of C 3c over C c and C c at the centreline for ˆb ij. It is interesting to observe that at the channel centerline the C c metric goes to zero. It is also interesting to note that the return to isotropy experiments for plane strain of Choi[5] when viewed in the barycentric map Fig. (4), (left) they show no tendency to get aligned with the axisymmetric expansion state as claimed by Choi et al.[5]. This fact is further substantiated by investigating the scalar metrics C c and C c in the D plot Fig. (4), (right) where C c signifies that return to isotropy from plane strain experiments cannot be claimed have the same behaviour like axi-symmetric expansion (C c = ). Figure shows how the barycentric map captures the channel ow in three-dimensional space[7]. This projection of a three-dimensional ow is possible only in linear domain and not in a non-linear domain such as the AIM. From Fig. 3 it can be observed that when a uniform distribution of randomly generated anisotropic stress tensors ˆb ij are plotted in the barycentric map and mapped back to the invariant map, the distribution is quite different. This fact illustrates that conclusion of physical facts (i.e nearness to axisymmetric and two-component limits) for any turbulent ow should not be made based on vi- C c,c c C c C c C 3c Lumley eigenvalue map Barycentric map y + Figure : At any point in the ow exact weighting can be found in terms of the limiting states. Data were extracted from a direct numerical simulation of fully developed plane turbulent channel ow for = 8 by Kim et al.[7] λ.5 Figure : Scalar metrics based on the eigenvalues of anisotropy tensor ˆb ij projected onto the barycentric map. Data were extracted from a direct numerical simulation of fully developed plane turbulent channel ow for = 8 by Kim et al.[7] III Figure 3: Randomly generated anisotropic tensors ˆbij which are uniformly distributed in the linear barycentric space (right) mapped back to the nonlinear space (III, II) (left). The nonlinearity in II, III of the invariant space concentrates the points near isotropy. The isotropic point corresponds to 3comp. 3 λ 4 comp isotropic comp ISSN: ISBN:

15 comp comp Proceedings of the 3th WSEAS International Conference on APPLIED MATHEMATICS (MATH'8) a ij C c,c c C 3c D 3c =. (88).3.. C c C c x/m Figure 4: Scalar metrics C c, C c and C 3c of the ˆb ij for plane strain (case plane strain[5]) plotted inside the barycentric map (left) and as metrics plot (right). sual observation. It has to be noted that invariant maps based on (II,III), (λ,λ ) and (C c,c c ) are not sensitive to the individual energy components of the anisotropic stress tensors, as it can be shown that two totally different individual energy component anisotropic stress tensors having trace zero will be mapped to the same point in the above anisotropy invariant maps. The information which is preserved between two anisotropic stress tensors mapped to two different points in the anisotropy invariant maps is their distance from the limiting states (one-component, two-component and three-component). For the one-component limit the invariants can be found as ( ) ( II = + ( + 3 3) 3) (89) III = ( ) 3 ( + 3 ( + 3 3) 3) 3. (9) For the two-component limit the invariants can be found as ( ) ( ) ( II = + + (9) 6 6 3) III = ( ) ( ) 3 ( + 3. (9) 6 3) For the isotropic limit the invariants can be found as II = (93). Relationship between Invariants and Barycentric Co-ordinates In this section, we prove that the axisymmetry and the two-dimensional relationships of the AIM can be derived from equations (57). The anisotropy tensor ˆb ij for the limiting states of turbulence are described in the following ways: ˆbc = ˆbc = ˆb3c = D c = D c traced c 3 D c traced c 3 D 3c traced 3c 3 D c = (83) (84) (85) (86) (87) III =. (94) The matrix for the axisymmetric expansion boundary from equation (57) and taking C c = is given as: /3 ˆbc 3c = C c /3 /3 + C 3c (95) Using relation (59), let us take C c = κ and C 3c = κ, therefore the invariants can be found as ( ) ( II = 3 κ + ) ( 3 κ + ) 3 κ (96) III = ( ) 3 ( 3 κ + ) 3 ( 3 κ + ) 3 3 κ (97) II = III = ( ) 6 κ (98) 9 ( ) κ 3. (99) 9 ISSN: ISBN: