Vector and Tensor Analysis in Turbomachinery Fluid Mechanics

Transcription

1 A Vector and Tensor Analysis in Turbomachinery Fluid Mechanics A. 1 Tensors in Three-Dimensional Euclidean Space In this section, we briefly introduce tensors, their significance to turbomachinery fluid dynamics and their applications. The tensor analysis is a powerful tool that enables the reader to study and to understand more effectively the fundamentals of fluid mechanics. Once the basics of tensor analysis are understood, the reader will be able to derive all conservation laws of fluid mechanics without memorizing any single equation. In this section, we focus on the tensor analytical application rather than mathematical details and proofs that are not primarily relevant to engineering students. To avoid unnecessary repetition, we present the definition of tensors from a unified point of view and use exclusively the three-dimensional Euclidean space, with N = 3 as the number of dimensions. The material presented in this chapter has drawn from classical tensor and vector analysis texts, among others those mentioned in References. It is tailored to specific needs of turbomchinery fluid mechanics and is considered to be helpful for readers with limited knowledge of tensor analysis. The quantities encountered in fluid dynamics are tensors. A physical quantity which has a definite magnitude but not a definite direction exhibits a zeroth-order tensor, which is a special category of tensors. In a N-dimensional Euclidean space, a zeroth-order tensor has N 0 = 1 component, which is basically its magnitude. In physical sciences, this category of tensors is well known as a scalar quantity, which has a definite magnitude but not a definite direction. Examples are: mass m, volume v, thermal energy Q (heat), mechanical energy W (work) and the entire thermo-fluid dynamic properties such as density, temperature T, enthalpy h, entropy s, etc. In contrast to the zeroth-order tensor, a first-order tensor encompasses physical quantities with a definite magnitude with components and a definite direction that can be decomposed in directions. This special category of tensors is known as vector. Distance X, velocity V, acceleration A, force F and moment of momentum M are few examples. A vector quantity is invariant with respect to a given category of coordinate systems. Changing the coordinate system by applying certain transformation rules, the vector components undergo certain changes resulting in a new set of components that are related, in a definite way, to the old ones. As we will see later, the order of the above tensors can be reduced if they are multiplied with each other in a scalar manner. The mechanical energy

2 686 A Vector and Tensor Analysis in Turbomachinery W = F.X is a representative example, that shows how a tensor order can be reduced. The reduction of order of tensors is called contraction. A second order tensor is a quantity, which has definite components and definite directions (in three-dimensional Euclidean space: ). General stress tensor, normal stress tensor, shear stress tensor, deformation tensor D and rotation tensor are few examples. Unlike the zeroth and first order tensors (scalars and vectors), the second and higher order tensors cannot be directly geometrically interpreted. However, they can easily be interpreted by looking at their pertinent force components, as seen later in section A.5.4 A.1.1 Index Notation In a three-dimensional Euclidean space, any arbitrary first order tensor or vector can be decomposed into 3 components. In a Cartesian coordinate system shown in Fig. A.1, the base vectors in x 1, x 2, x 3 directions e 1, e 2, e 3 are perpendicular to each other and have the magnitude of unity, therefore, they are called orthonormal unit vectors. Furthermore, these base vectors are not dependent upon the coordinates, therefore, their derivatives with respect to any coordinates are identically zero. In contrast, in a general curvilinear coordinate system (discussed in Appendix A) the base vectors Fig. A.1: Vector decomposition in a Cartesian coordinate system

3 A Vector and Tensor Analysis in Turbomachinery 687 do not have the magnitude of unity. They depend on the curvilinear coordinates, thus, their derivatives with respect to the coordinates do not vanish. As an example, vector A with its components A 1, A 2 and A 3 in a Cartesian coordinate system shown in Fig. A.1 is written as: According to Einstein's summation convention, it can be written as: (A.1) (A.2) The above form is called the index notation. Whenever the same index (in the above equation i) appears twice, the summation is carried out from 1 to N (N = 3 for Euclidean space). A.2 Vector Operations: Scalar, Vector and Tensor Products A.2.1 Scalar Product Scalar or dot product of two vectors results in a scalar quantity. We apply the Einstein s summation convention defined in Eq.(A.2) to the above vectors: (A.3) we rearrange the unit vectors and the components separately: (A.4) In Cartesian coordinate system, the scalar product of two unit vectors is called Kronecker delta, which is: with ij as Kronecker delta. Using the Kronecker delta, we get: (A.5) (A.6) The non-zero components are found only for i = j, or ij = 1, which means in the above equation the index j must be replaced by i resulting in: with scalar C as the result of scalar multiplication. (A.7)

4 688 A Vector and Tensor Analysis in Turbomachinery A.2.2 Vector or Cross Product The vector product of two vectors is a vector that is perpendicular to the plane described by those two vectors. Example: with C as the resulting vector. We apply the index notation to Eq. (A.8): (A.8) (A.9) with ijk as the permutation symbol with the following definition illustrated in Fig. A.2: ijk = 0 for i = j, j = k or i = k: (e.g. 122) ijk = 1 for cyclic permutation: (e.g. 123) ijk = -1 for anti-cyclic permutation: (e.g.132) Fig. A.2: Permutation symbol, (a) positive, (b) negative permutation Using the above definition, the vector product is given by: (A.10) A.2.3 Tensor Product The tensor product is a product of two or more vectors, where the unit vectors are not subject to scalar or vector operation. Consider the following tensor operation: (A.11)

5 A Vector and Tensor Analysis in Turbomachinery 689 The result of this purely mathematical operation is a second order tensor with nine components: (A.12) The operation with any tensor such as the above second order one acquires a physical meaning if it is multiplied with a vector (or another tensor) in scalar manner. Consider the scalar product of the vector C and the second order tensor. The result of this operation is a first order tensor or a vector. The following example should clarify this: Rearranging the unit vectors and the components separately: (A.13) (A.14) It should be pointed out that in the above equation, the unit vector e k must be multiplied with the closest unit vector namely e i (A.15) The result of this tensor operation is a vector with the same direction as vector B. Different results are obtained if the positions of the terms in a dot product of a vector with a tensor are reversed as shown in the following operation: The result of this operation is a vector in direction of A. Thus, the products is different from (A.16) A.3 Contraction of Tensors As shown above, the scalar product of a second order tensor with a first order one is a first order tensor or a vector. This operation is called contraction. The trace of a second order tensor is a tensor of zero th order, which is a result of a contraction and is a scalar quantity. (A.17) As can be shown easily, the trace of a second order tensor is the sum of the diagonal element of the matrix ij. If the tensor itself is the result of a contraction of two second order tensors and D:

6 690 A Vector and Tensor Analysis in Turbomachinery (A.18) then the Tr( ) is: (A.19) A.4 Differential Operators in Fluid Mechanics In fluid mechanics, the particles of the working medium undergo a time dependent or unsteady motion. The flow quantities such as the velocity V and the thermodynamic properties of the working substance such as pressure p, temperature T, density or any arbitrary flow quantity Q are generally functions of space and time: During the flow process, these quantities generally change with respect to time and space. The following operators account for the substantial, spatial, and temporal changes of the flow quantities. A.4.1 Substantial Derivatives The temporal and spatial change of the above quantities is described most appropriately by the substantial or material derivative. Generally, the substantial derivative of a flow quantity Q, which may be a scalar, a vector or a tensor valued function, is given by: (A.20) The operator D represents the substantial or material change of the quantity Q, the first term on the right hand side of Eq. (A.20) represents the local or temporal change of the quantity Q with respect to a fixed position vector x. The operator d symbolizes the spatial or convective change of the same quantity with respect to a fixed instant of time. The convective change of Q may be expressed as: A simple rearrangement of the above equation results in: (A.21) (A.22) The scalar multiplication of the expressions in the two parentheses of Eq. (A.22) results in Eq. (A.21)

7 A.4.2 Differential Operator A Vector and Tensor Analysis in Turbomachinery 691 The expression in the second parenthesis of Eq. (A.22) is the spatial differential operator (Nabla, del) which has a vector character. In Cartesian coordinate system, the operator Nabla is defined as: Using the above differential operator, the change of the quantity Q is written as: (A.23) (A.24) The increment dq in Eq. (A.24) is obtained either by applying the product, or by taking the dot product of the vector dx and Q. If Q is a scalar quantity, then Q is a vector or a first order tensor with definite components. In this case, Q is called Fig. A.3: Physical explanation of the gradient of scalar field

8 692 A Vector and Tensor Analysis in Turbomachinery the gradient of the scalar field. Equation (A.24) indicates that the spatial change of the quantity Q assumes a maximum if the vector Q (gradient of Q ) is parallel to the vector dx. If the vector Q is perpendicular to the vector dx, their product will be zero. This is only possible, if the spatial change dx occurs on a surface with Q = const. Consequently, the quantity Q does not experience any changes. The physical interpretation of this statement is found in Fig. A.3. The scalar field is represented by the point function temperature that changes from the surfaces T to the surface T + dt. In Fig. A.3, the gradient of the temperature field is shown as T, which is perpendicular to the surface T = const. at point P. The temperature probe located at P moves on the surface T = const. to the point M, thus measuring no changes in temperature ( = /2, cos = 0). However, the same probe experiences a certain change in temperature by moving to the point Q, which is characterized by a higher temperature T + dt ( 0 < < /2 ). The change dt can immediately be measured, if the probe is moved parallel to the vector T. In this case, the displacement dx (see Fig. A.3) is the shortest ( = 0, cos = 1). Performing the similar operation for a vector quantity as seen in Eq.(A.21) yields: The right-hand side of Eq. (A.25) is identical with: (A.25) (A.26) In Eq. (A.26) the product can be considered as an operator that is applied to the vector V resulting in an increment of the velocity vector. Performing the scalar multiplication between dx and gives: (A.27) with Vas the gradient of the vector field which is a second order tensor. To perform the differential operation, first the operator is applied to the vector V, resulting in a second order tensor. This tensor is then multiplied with the vector dx in a scalar manner that results in a first order tensor or a vector. From this operation, it follows that the spatial change of the velocity vector can be expressed as the scalar product of the vector dx and the second order tensor V, which represents the spatial gradient of the velocity vector. Using the spatial derivative from Eq. (A.27), the substantial change of the velocity is obtained by: where the spatial change of the velocity is expressed as : Dividing Eq. (A.29) by dt yields the convective part of the acceleration vector: (A.28) (A.29)

9 A Vector and Tensor Analysis in Turbomachinery 693 The substantial acceleration is then: (A.30) (A.31) The differential dt may symbolically be replaced by Dt indicating the material character of the derivatives. Applying the index notation to velocity vector and Nabla operator, performing the vector operation, and using the Kronecker delta, the index notation of the material acceleration A is: (A.32) Equation (A.32) is valid only for Cartesian coordinate system, where the unit vectors do not depend upon the coordinates and are constant. Thus, their derivatives with respect to the coordinates disappear identically. To arrive at Eq. (A.32) with a unified index i, we renamed the indices. To decompose the above acceleration vector into three components, we cancel the unit vector from both side in Eq. (A.32) and get: (A.33) To find the components in x i -direction, the index i assumes subsequently the values from 1 to 3, while the summation convention is applied to the free index j. As a result we obtain the three components: (A.34) A.5 Operator Applied to Different Functions This section summarizes the applications of nabla operator to different functions. As mentioned previously, the spatial differential operator has a vector character. If it acts on a scalar function, such as temperature, pressure, enthalpy etc., the result is a vector and is called the gradient of the corresponding scalar field, such as gradient of temperature, pressure, etc. (see also previous discussion of the physical interpretation of Q). If, on the other hand, acts on a vector, three different cases are distinguished.

10 694 A Vector and Tensor Analysis in Turbomachinery A.5.1 Scalar Product of and V This operation is called the divergence of the vector V. The result is a zero th -order tensor or a scalar quantity. Using the index notation, the divergence of V is written as: (A.35) The physical interpretation of this purely mathematical operation is shown in Fig. A.4. The mass flow balance for a steady incompressible flow through an infinitesimal volume dv = dx 1 dx 2 dx 3 is shown in Fig. A.4. V in V 2 V 2 x 2 dx 2 x 2 n S V V 3 ds V V 1 V 1 1 dx 1 n ds V 2 x 1 V 3 V 3 3 dx 3 S x 3 V out n out Fig. A.4: Physical interpretation of.v We first establish the entering and exiting mass flows through the cube side areas perpendicular to x 1 - direction given by da 1 = dx 2 dx 3 : (A.36) (A.37) Repeating the same procedure for the cube side areas perpendicular to x 2 and x 3 directions given by da 2 = dx 3 dx 1 and da 3 = dx 1 dx 2 and subtracting the entering mass flows from the exiting ones, we obtain the net mass net flow balances through the infinitesimal differential volume as: (A.38)

11 A Vector and Tensor Analysis in Turbomachinery 695 The right hand side of Equation (A.38) is a product of three terms, the density, the differential volume dv and the divergence of the vector V. Since the first two terms are not zero, the divergence of the vector must disappear. As result, we find: (A.39) Equation (A.39) expresses the continuity equation for an incompressible flow, as we will saw in Chapter 3. A.5.2 Vector Product This operation is called the rotation or curl of the velocity vector V. Its result is a first-order tensor or a vector quantity. Using the index notation, the curl of V is written as: (A.40) The curl of the velocity vector is known as vorticity,. As we will see later, the vorticity plays a crucial role in fluid mechanics. It is a characteristic of a rotational flow. For viscous flows encountered in engineering applications, the curl is always different from zero. To simplify the flow situation and to solve the equation of motion, as we discusseded in Chapter 3 later, the vorticity vector, can under certain conditions, be set equal to zero. This special case is called the irrotational flow. A.5.3 Tensor Product of and V This operation is called the gradient of the velocity vector V. Its result is a second tensor. Using the index notation, the gradient of the vector V is written as: (A.41) Equation (A.41) is a second order tensor with nine components and describes the deformation and the rotation kinematics of the fluid particle. As we saw previously, the scalar multiplication of this tensor with the velocity vector, resulted in the convective part of the acceleration vector, Eq. (A.32). In addition to the applications we discussed, can be applied to a product of two or more vectors by using the Leibnitz's chain rule of differentiation: (A.42) For U = V, Eq. (A.42) becomes or (A.43)

12 696 A Vector and Tensor Analysis in Turbomachinery Equation (A.43) is used to express the convective part of the acceleration in terms of the gradient of kinetic energy of the flow. A.5.4 Scalar Product of and a Second Order Tensor Consider a fluid element with sides dx 1, dx 2, dx 3 parallel to the axis of a Cartesian coordinate system, Fig.A.5. The fluid element is under a general three-dimensional stress condition. The force vectors acting on the surfaces, which are perpendicular to the coordinates x 1, x 2, and x 3 are denoted by F 1, F 2, and F 3,. The opposite surfaces are subject to forces that have experienced infinitesimal changes F 1 + df 1, F 2 + df 2, and F 3 + df 3. Each of these force vectors is decomposed into three components F ij according to the coordinate system defined in Fig. A.5. The first index i refers to the axis, to which the fluid element surface is perpendicular, whereas the second index j indicates the direction of the force component. We divide the individual components of the above force vectors by their corresponding area of the fluid element side. The results of these divisions exhibit the components of a second order stress tensor represented by as shown in Fig. A.6. Fig. A.5: Fluid element under a general three-dimensional stress condition

13 A Vector and Tensor Analysis in Turbomachinery 697 Fig. A.6: General stress condition As an example, we take the force component and divide it by the corresponding area results in. Correspondingly, we divide the force component on the opposite surface by the same area and obtain. In a similar way we find the remaining stress components, which are shown in Fig. A.6. The tensor has nine components ij as the result of forces that are acting on surfaces. Similar to the force components, the first index i refers to the axis, to which the fluid element surface is perpendicular, whereas the second index j indicates the direction of the stress component. Considering the stress situation in Fig.A.6, we are now interested in finding the resultant force acting on the fluid particle that occupies the volume element. For this purpose, we look at the two opposite surfaces that are perpendicular to the axis as shown in Fig. A.7.

14 698 A Vector and Tensor Analysis in Turbomachinery Fig. A.7: Stresses on two opposite walls As this figure shows, we are dealing with 3 stress components on each surface, from which one on each side is the normal stress component such as and. The remaining components are the shear stress components such as and. According to Fig. A.7 the force balance in -directions is: and in -direction, we find

15 A Vector and Tensor Analysis in Turbomachinery 699 Similarly, in, we obtain Thus, the resultant force acting on these two opposite surfaces is: In a similar way, we find the forces acting on the other four surfaces. The total resulting forces acting on the entire surface of the element are obtained by adding the nine components. Defining the volume element, we divide the results by and obtain the resulting force vector that is acting on the volume element. (A.44) Since the stress tensor is written as: it can be easily shown that: (A.45) (A.46) The expression is a scalar differentiation of the second order stress tensor and is called the divergence of the tensor field. We conclude that the force vector acting on the surface of a fluid element is the divergence of its stress tensor. The stress tensor is usually divided into its normal- and shear stress parts. For an incompressible fluid it can be written as

16 700 A Vector and Tensor Analysis in Turbomachinery (A.47) with Ip as the normal and T as the shear stress tensor. The normal stress tensor is a product of the unit tensor and the pressure p. Inserting Eq. (A.47) into (A.46) leads to (A.48) Its components are (A.49) A.5.5 Eigenvalue and Eigenvector of a Second Order Tensor The velocity gradient expressed in Eq. (A.41) can be decomposed into a symmetric deformation tensor D and an antisymmetric rotation tensor : (A.50) A scalar multiplication of D with any arbitrary vector A may result in a vector, which has an arbitrary direction. However, there exists a particular vector V such that its scalar multiplication with D results in a vector, which is parallel to V but has different magnitude: (A.51) with V as the eigenvector and the eigenvalue of the second order tensor D. Since any vector can be expressed as a scalar product of the unit tensor and the vector itself, we may write: (A.52) Equation (A.52) can be rearranged as: (A.53) and its index notation gives: (A.54) or

17 A Vector and Tensor Analysis in Turbomachinery 701 Expanding Eq. (A.55) gives a system of linear equations, (A.55) (A.56) A nontrivial solution of these Eq. (A.56) is possible if and only if the following determinant vanishes: or in index notation: Expanding Eq. (A.58) results in (A.57) (A.58) (A.59) After expanding the above determinant, we obtain an algebraic equation in in the following form where and are invariants of the second order tensor D defined as: (A.60) (A.61) (A.62) (A.63) The roots of Eq. (A.60) are known as the eigenvalues of the tensor D. The superscript refers to the roots of Eq. (A.60) - not to be confused with the component of a vector.

18 702 A Vector and Tensor Analysis in Turbomachinery References 1. Aris, R.: Vector, Tensors and the Basic Equations of Fluid Mechanics. Prentice- Hall, Englewood Cliffs (1962) 2. Brand, L.: Vector and Tensor Analysis. John Wiley and Sons, New York (1947) 3. Klingbeil, E.: Tensorrechnung für Ingenieure. Bibliographisches Institut, Mannheim (1966) 4. Lagally, M.: Vorlesung über Vektorrechnung, 3rd edn. Akademische Verlagsgesellschaft, Leipzig (1944) 5. Vavra, M.H.: Aero-Thermodynamics and Flow in Turbomachines. John Wiley & Sons, Chichester (1960)

19 B Tensor Operations in Orthogonal Curvilinear Coordinate Systems B.1 Change of Coordinate System The vector and tensor operations we have discussed in the foregoing chapters were performed solely in rectangular coordinate system. It should be pointed out that we were dealing with quantities such as velocity, acceleration, and pressure gradient that are independent of any coordinate system within a certain frame of reference. In this connection it is necessary to distinguish between a coordinate system and a frame of reference. The following example should clarify this distinction. In an absolute frame of reference, the flow velocity vector may be described by the rectangular Cartesian coordinate x i : (B.1) It may also be described by a cylindrical coordinate system, which is a non-cartesian coordinate system: (B.2) or generally by any other non-cartesian or curvilinear coordinate system i that describes the flow channel geometry: (B.3) By changing the coordinate system, the flow velocity vector will not change. It remains invariant under any transformation of coordinates. This is true for any other quantities such as acceleration, force, pressure- or temperature gradient. The concept of invariance, however, is generally no longer valid if we change the frame of reference. If, for example, the flow particles leave the absolute frame of reference (such as turbine or compressor stator blade channels) and enter a relative frame, for example rotating turbine or compressor blade channels, its velocity will experience a change as seen in Chapter 3. In this Chapter, we will pursue the concept of quantity invariance and discuss the fundamentals needed for understanding the coordinate transformation.

20 704 B Tensor Operations in Orthogonal Curvilinear Coordinate Systems B.2 Co- and Contravariant Base Vectors, Metric Coefficients As we saw in Appendix A, a vector quantity is described in Cartesian coordinate system x i by its components: (B.4) with e i as orthonormal unit vectors (Fig. B.1 left). Fig. B.1: Base vectors in a Cartesian (left ) and in a generalized orthogonal curvilinear coordinate system (right) The same vector transformed into the curvilinear coordinate system k (Fig. B.1 right) is represented by: (B.5) where g k are termed covariant base vectors and V k the contravariant components of V with respect to the base g k. For curvilinear coordinate systems, we place the indices diagonally for summing convenience. Unlike the Cartesian base vectors e i, that are orthonormal vectors (of unit length and mutually orthogonal), the base vectors g k do not have unit lengths. The base vectors g k represent the rate of change of the position vector x with respect to the curvilinear coordinates i. (B.6) Since in a Cartesian coordinate system the unit vectors e i, are not functions of the coordinates x i, Eq. (B.6) can be written as:

21 B Tensor Operations in Orthogonal Curvilinear Coordinate Systems 705 Fig. B.2: Co- and contravariant base vectors Similarly, the reciprocal or contravariant base vector g k is defined as: (B.7) (B.8) As shown in Fig. B.2, the covariant base vectors g 2, g 2, and g 3 are tangent vectors to the mutually orthogonal curvilinear coordinates 1, 2, and 3. The reciprocal base vectors g 1, g 2, g 3, however, are orthogonal to the planes described by g 2 and g 3, g 3 and g 1, and g 1 and g 2, respectively. These base vectors are interrelated by: (B.9) where g k and g j are referred to as the covariant and contravariant base vectors, respectively. The new Kronecker delta k j from Eq. (B.9) has the values: The vector V written relative to its co- and contravariant base is: (B.10)

22 706 B Tensor Operations in Orthogonal Curvilinear Coordinate Systems with the components V k and V k as the covariant and contravariant components, respectively. The scalar product of covariant respectively contravariant base vectors results in the covariant and covariant metric coefficients: The mixed metric coefficient is defined as (B.11) (B.12) Using in curvilinear coordinate systems, its is often necessary to express the covariant base vectors in terms of the contravariant ones and vice versa. To do this, we first assume that: (B.13) Generally the contravariant base vector can be written as (B.14) To find a direct relation between the base vectors, first the coefficient matrix A ij must be determined. To do so, we multiply Eq. (B.14) with g k scalarly: (B.15) This leads to. The right hand side is different from zero only if j = k, which means: (B.16) Introducing Eq. (B.16) into (B.14) results in a relation that expresses the contravariant base vectors in terms of covariant base vectors: (B.17) The covariant base vector can also be expressed in terms of contravariant base vectors in a similar way: (B.18) Multiply Eq. (B.l8) with (B.17) establishes a relationship between the covariant and contravariant metric coefficients: Applying the Kronecker delta on the right hand side results in: (B.19) (B.20)

23 B Tensor Operations in Orthogonal Curvilinear Coordinate Systems 707 B.3 Physical Components of a Vector As mentioned previously, the base vectors g i or g j are not unit vectors. Consequently the co- and contravariant vector components V j or V j do not reflect the physical components of vector V. To obtain the physical components, first the corresponding unit vectors must be found. The covariant unit vectors are determined from: Similarly, the contravariant unit vectors are: (B.21) (B.22) where g i*, represents the unit base vector, g i the absolute value of the base vector. The expression (ii) denotes that no summing is carried out, whenever the indices are enclosed within parentheses. The vector can now be expressed in terms of its unit base vectors and the corresponding physical components: (B.23) Thus the covariant and contravariant physical components can be easily obtained from: B.4 Derivatives of the Base Vectors, Christoffel Symbols (B.24) In a curvilinear coordinate system, the base vectors are generally functions of the coordinates itself. This fact must be considered while differentiating the base vectors. Consider the derivative: (B.25) The comma in Eq. (B.25) followed by the lower index j represents the derivative of the covariant base vector g i with respect to the coordinate j. Similar to Eq.(B.7), the unit vector e k can be written as: Introducing Eq. (B.26) into (B.25) yields: (B.26) (B.27) with ijn, and i j n as the Christoffel symbol of first and second kind, respectively with the definitions:

24 708 B Tensor Operations in Orthogonal Curvilinear Coordinate Systems (B.28a) It follows from (B.28a) that the Christoffel symbols of the second kind are related to the first kind by: (B.28b) Because of (B.28b) and, for the sake of simplicity, in what follows, we will use the Christoffel symbol of second kind. Thus, the derivative of the contravariant base vector reads: The Christoffel symbols are then obtained by expanding Eq. (B.28a): (B.29) (B.30a) The fact that the only non-zero elements of the matrix allows the modification of Eq. (B.30a) to: are the diagonal elements (B.30b) In Eq. (B.30a), the Christoffel symbols are symmetric in their lower indices. Again, note that a repeated index in parentheses in an expression such as g (kk) does not subject to summation. B.5 Spatial Derivatives in Curvilinear Coordinate System The differential operator is in curvilinear coordinate system defined as: (B.31) In this Chapter, the operator will be applied to different arguments such as zeroth, first and second order tensors. B.5.1 Application of to Zeroth Order Tensor Functions If the argument is a zeroth order tensor which is a scalar quantity such as pressure or temperature, the results of the operation is the gradient of the scalar field which is a vector quantity:

25 B Tensor Operations in Orthogonal Curvilinear Coordinate Systems 709 (B.32) The abbreviation,i refers to the derivative of the argument, in this case p, with respect to the coordinate i. B.5.2 Application of to First and Second Order Tensor Functions Scalar Product: If the argument is a first order tensor such as a velocity vector, the order of the resulting tensor depends on the operation character between the operator and the argument. If the argument is a product of two or more quantities, chain rule of differentiation is applied. Thus, the divergence of the vector filed V is: (B.33a) Implementing the Christoffel symbol from (B.27), the results of the above operations are the divergence of the vector V. It should be noticed that a scalar operation leads to a contraction of the order of tensor on which the operator is acting. The scalar operation in (B.33a) leads to: (B.33b) Cross Product: This vector operation yields the rotation or curl of a vector field as: (B.34a) Because of the antisymmetric character of a cross product and the symmetry of the Christoffel symbols in their lower indices, the second term on the right-hand side of (B.34a), namely. As a result, we find (B.34b) with as the permutation symbol that functions similar to the one for Cartesian coordinate system withe the difference that:

26 710 B Tensor Operations in Orthogonal Curvilinear Coordinate Systems (B.35) with as the determinant of the matrix of covariant metric coefficients. Tensor Product: The gradient of a first order tensor such as the velocity vector V is a second order tensor. Its index notation in a curvilinear coordinate system is: (B.36) Divergence of a Second Order Tensor: A scalar operation that involves and a second order tensor, such as the stress tensor or deformation tensor D, results in a first order tensor which is a vector: (B.37) The right hand side of (B.37) is reduced to: (B.38) By calculating the shear forces using the Navier-Stokes equation, the second derivative, the Laplace operator, is needed: (B.39) This operator applied to the velocity vector yields: (B.40) B.6 Application Example 1: Inviscid Incompressible Flow Motion As the first application example, the equation of motion for a steady, incompressible and inviscid flow is transformed into a cylindrical coordinate system, where it is decomposed into components in r, and z-directions. Neglecting the gravitational force, the coordinate invariant version of the equation is written as:

27 B Tensor Operations in Orthogonal Curvilinear Coordinate Systems 711 (B.41) The transformation and decomposition procedure is shown in the following steps. B.6.1 Equation of Motion in Curvilinear Coordinate Systems The second order tensor on the left hand side can be obtained using Eq. (B.35): (B.42) The scalar multiplication of (B.42) with the velocity vector V leads to: (B.43) Introducing the mixed Kronecker delta into (B.43), we get: (B.44) For an orthogonal curvilinear coordinate system the mixed Kronecker delta is: (B.45) Taking this into account, Eq. (B.43) (B.46) and rearranging the indices, we obtain the convective part of the Navier-Stokes equations as: (B.47) The pressure gradient on the right hand side of Eq. (B.40) is calculated form Eq. (B.32): (B.48) Replacing the contravariant base vector with the covariant one using Eq. (B.17) leads to: (B.49)

28 712 B Tensor Operations in Orthogonal Curvilinear Coordinate Systems Incorporating Eqs. (B.46) and (B.49) into Eq. (B.40) yields: (B.50) In i-direction, the equation of motion is: (B.51) B.6.2 Special Case: Cylindrical Coordinate System To transfer Eq. (B.40) into any arbitrary orthogonal curvilinear coordinate system, first the coordinate system must be specified. The cylindrical coordinate system is related to the Cartesian coordinate system by: The curvilinear coordinate system is represented by: (B.52) (B.53) B.6.3 Base Vectors, Metric Coefficients The base vectors are calculated from Eq. (B.7). (B.54) Equation (B.54) decomposed in its components yields: (B.55) Performing the differentiation of the terms in Eq. (B.55) yields: (B.56)

29 B Tensor Operations in Orthogonal Curvilinear Coordinate Systems 713 The co- and contravariant metric coefficients are: (B.57) The contravariant base vectors are obtained from: (A.57a) Since the mixed metric coefficient are zero, (B.57a) reduces to: (A.57b) B.6.4 Christoffel Symbols The Christoffel symbols are calculated from Eq. (B.30b) (B.59) To follow the calculation procedure, one zero- element and one non-zero element are calculated: (B.60) All other elements are calculated similarly. They are shown in the following matrices: (B.61)

30 714 B Tensor Operations in Orthogonal Curvilinear Coordinate Systems Introducing the non-zero Christoffel symbols into Eq. (B.50), the components in g l, g 2, and g 3 directions are: (B.62) (B.64) (B.63) B.6.5 Introduction of Physical Components The physical components can be calculated from Eqs. (B.21) and (B.24): (B.65) The V i -components expressed in terms of V *i are: (B.66) Introducing Eqs.(B.65) into (B.61), (B.62), and (B.63) results in: (B.67) (B.68) (B.69) According to the definition: the physical components of the velocity vectors are: (B.70)

31 B Tensor Operations in Orthogonal Curvilinear Coordinate Systems 715 (B.71) and insert these relations into Eqs. (B.66) to (B.68), the resulting components in r,, and z directions are: (B.72) B.7 Application Example 2: Viscous Flow Motion As the second application example, the Navier-Stokes equation of motion for a steady, incompressible flow with constant viscosity is transferred into a cylindrical coordinate system, where it is decomposed into its three components r,, z. The coordinate invariant version of the equation is written as: (B.73) The second term on the right hand side of Eq. (B.73) exhibits the shear stress force. It was treated in section B.5, Eq. (B.39) and is the only term that has been added to the equation of motion for inviscid flow, Eq. (B.40). B.7.1 Equation of Motion in Curvilinear Coordinate Systems The transformation and decomposition procedure is similar to the example in section B. 6. Therefore, a step by step derivation is not necessary. (B.74)

32 716 B Tensor Operations in Orthogonal Curvilinear Coordinate Systems B.7.2 Special Case: Cylindrical Coordinate System Using the Christoffel symbols from section B.6.4 and the physical components from B.6.5, and inserting the corresponding relations these relations into Eq. (B.74), the resulting components in r,, and z directions are: (B.75) (B.76) References 1. Aris, R.: Vector, Tensors and the Basic Equations of Fluid Mechanics. Prentice- Hall, Inc., Englewood Cliffs (1962) 2. Brand, L.: Vector and Tensor Analysis. John Wiley & Sons, New York (1947) 3. Klingbeil, E.: Tensorrechnung für Ingenieure. Bibliographisches Institut, Mannheim (1966) 4. Lagally, M.: Vorlesung über Vektorrechnung, 3rd edn. Akademische Verlagsgesellschaft, Leipzig (1944) 5. Vavra, M.H.: Aero-Thermodynamics and Flow in Turbomachines. John Wiley & Sons, Chichester (1960)

33 Index 1/7-th boundary layer displacement thickness 631 energy deficiency thickness 631, 633 integral equation 635 length scale 670 momentum deficiency thickness 631 re-attachment 661, 664, 667, 668, 670 separation 667 wall influence 655 Algebraic Model 585 Baldwin-Lomax 585 Cebeci-Smith 584 Prandtl Mixing Length 578 Anemometer 519 Averaging 560 conservation equations 561 continuity equation 561 mechanical energy equation 562 Navier-stokes equation 561 total enthalpy equation 565 Axial vector 24 Base vectors 704, 707 contravariant 704 covariant 704 reciprocal 704 Becalming 535 Bernoulli equation 39, 584 Bio-Savart law 195 Blade design 267 Assessment of 289 Bézier 285 camberline coordinates 285 conformal transformation 267 Blade design using Bézier function 283 Blade force, inviscid flow circulation , , 168, 171 lift coefficient 149, 153, , 166, 168 lift force , 153 Blade force, viscous flow drag coefficient 153, 157 drag force 152, 153 drag-lift ratio 157, 158 lift coefficient 149, 153, , 166, 168 lift force , 153 lift-solidity coefficient , , no-slip condition 150 optimum solidity 158, 160 periodic wake flow 150 total pressure loss coefficient 150 Blending function 590 Boundary layer 580 buffer layer 580 bypass 517 experimental simulation of unsteady 527 intermittency function 519 logarithmic layer 580 natural 535 outer layer 581 transitional 581 unsteady Boundary 525 viscous sublayer 580 von Karman constant 583

34 718 Index wake function 581 wake induced 535 Boundary layer theory Blasius solution 624 concept of 619 viscous layer 619 Boussinesq 577 relationship 577 Bypass transition 517 Calmed region 532 Cascade process 546 Cauchy-Poisson law 35 Circulation function 389 Co- and contravariant Base Vectors 703 Combustion chamber 80, 84, 341, 346, 348, 372 chamber heat transfer 376 mass flow transients 373 temperature transients 374 Compound lean , Compression waves 100 Compressor corrections blade thickness effect 408 circulation function , 408 Mach number effect 394, 407 maximum velocity 386, , 393, 395 optimum flow condition 389 profile losses 385, 396, 407, 408 Reynolds number effect 408 secondary flow losses 385, 403, 406 shock losses 384, 385, 397, 400, 403 Compressor blade design 2, 265, 272, 273, 275, 279 base profile 2, 265, , 281, 285, 286 DCA, MCA 2, 267, 272, 273, 279, 281 double circular arc (DCA) 272 for High Subsonic Mach Number 279 graphic design of camberline 284 induced velocity 274, 275 Joukowsky transformation 267 maximum thickness 273, 277, 278, 285 multi-circular arc (MCA) profiles 272 NACA-67 profiles 2, 272, 274 subsonic compressor 271, 273, 279 summary, simple compressor blade design steps 279 superimposing the base profile 276, 278 supersonic compressor 282 vortex distribution 274 Compressor diffusion factor compressibility effect on, 393, 395, 396, 403 diffusion factor , , , , 415, 416, 420, 422 modified diffusion factor 385, 391, , 415, 416 Compressor stage 341, 348, 353 Conformal transformation circle-cambered Airfoil Transformation 271 circle-ellipse Transformation 268 circle-symmetric Airfoil Transformation 269 Convergent nozzle 91, 92 Cooled turbine 60, 82 Coriolis acceleration 51 Correlations autocorrelation 549 coefficient 548 osculating parabola 553 single point 548 tensor 548 two-point correlation 548 Covariance 549 Critical state 88, 97, 103 Cross-section change 93 Curvilinear coordinate Systems 703, 712 base vectors 712 metric coefficients 712 spatial Derivatives 708

35 Index 719 Deformation 34, 700 Density change 91 Derivative 690 material 690 substantial 690 temporal 690 Description 15 Lagrangian description 16 material 15 spatial 15 Diabatic expansion 351 Diabatic processes 351 Diabatic turbine module 467 Differential operator 691 Differential operators 690 Diffusers 220 concave and convex 225 configurations 221 conversion coefficient 224 induced Velocity 194 opening angle 226 pressure recovery 220, 222 recovery factor 224 vortex generator edges 226 Diffusivity 546 Direct Navier-Stokes simulations 577 Dissipation 575, 576, 577 dissipation 564 energy 545 equation 554 exact derivation of 577 function 43 kinetic Energy 576 parameter 557 range 556 turbulence 554 turbulent 564 viscous 564 Eddy viscosity 578 Effect of stage parameters on degree of reaction 130, 131, , 137, 139, 140 flow coefficient 129, 137, 139, 140, 142 load coefficient 129, 130, Efficiency 452 Efficiency of multi-stage turbomachines 233 heat recovery 233, 234, 237, 239 infinitesimally small expansion 235 isentropic efficiency polytropic efficiency 233, 235, recovery factor reheat 233, 234, 240, 241 reheat factor 240, 241 Einstein summation convention 37 Einstein s summation 687 Energy balance in stationary frame 42 dissipation function 43 mechanical energy 42 thermal energy balance 45 Energy cascade process 546, 547 Energy spectral function 558 Energy spectrum 555 dissipation range 556 large eddies 555 Energy transfer compressor stage 123, 125, , 137, 139, 143 Euler turbine equation 127 turbine row 127, 128, 136 turbine stage 124, 127, 128, , 139 Ensemble averaging 530 Entropy balance 49 Equation 570 turbulence kinetic Energy 570 Euler equation of motion 38 Expansion waves 98, 100, 121 Expansion adiabatic 348 Falkner-Skan equation 629 Fanno 101, Fanno line 101, Faulkner-Skan equation 628 Flow coefficients 453 Fluctuation kinetic energy 566 Frame indifferent quantities 34 Free turbulent flow 545 Friction stress tensor 35 Frozen turbulence 551

36 720 Index Gas dynamic functions 97 Gas turbines components 473 compressed air storage 473 design 473 dynamic 473 multi-spool 475 performance 473 power generation 473 process 473 shutdown 495 single spool 500 startup 495 three-spool 493 twin-spool 473 ultra high efficiency 485 Gaussian distribution 534 Generalized lift-solidity coefficient 162, 166 lift-solidity coefficient , , turbine rotor 163, 168, 170 turbine stator 159, , 166 Generic component configuration adiabatic compressor 348 adiabatic turbine 349, 350 compressor 341, 342, 346, 348, diabatic turbine 350 heat exchangers 346 inlet, exhaust, pipe 345 schematic 350, 353, 355 turbine , Generic modeling components 2 heat exchangers 346 inlet, exhaust, pipe 345 Geometry parameter 458 Heat transfer 376 conduction 376 convection 373 Nusselt number 660 radiation 376 Stanton number 660 thermography 660 Hot wire anemometry 653 aliasing effect 656 analog/digital converter 656 constant current (CC) mode 654 constant-temperature (CT) mode 654 cross-wire 654 folding frequency 656 Nyquist-frequency 656 sample frequency 656 sampling rate 656 signal conditioner 656 single wire 653 single, cross and three-wire probes 653 single-wire 653 three-wire 654 Incidence and deviation 241 cascade parameters 255 complex plane 242 complex velocity potential 243 conformal transformation 241, 242, 244, 250, 261 deviation 241, 255, 256, , 263 infinitely thin circular arc 250 optimum incidence 256, 257, 259 singularities 244, 247 transformation function 244, 247 Incompressibility condition 31 Index Notation 686, 687 Integral balances 59 axial moment 70 energy 72, 73, 75, 77, 78 linear momentum 59, 61 mass flow 60 moment of momentum 66, 68, 70, 71, 73 reaction force 65, 66, 68 shaft power 75, 78, 101 Intermittency 533 averaged 534 curved plate 521 ensemble-averaged 533 flat plate 524 function 653 Intermittency into Navier Stokes Equations 536

37 Index 721 maximum 533 minimum 534 modeling 524 Irreversibility 72, 79, 84 total pressure loss 79, 86, 87 Isentropic row load coefficients 452 Isotropic turbulence 560 Isotropy 547 Jacobian 16 21, 25, 26 function 16 19, 21, 25, 26 functional determinant 19 material derivative 17, 19, 26 transformation vector 16, 17, 18,19 21, 25 Joukowsky transformation 265 Kinematics of fluid motion 15 deformation tensor 21 material volume 17, 18, 20, 25 rotation tensor 24, 25 rotation vector 24, 25 translation 21, 23 Kinetic energy 556, 560, 566 Kolmogorov 546 eddies 546 first hypothesis 556 hypothesis 546 inertial subrange 555, 556 length scale 555 scales 555 second hypothesis 557 time scale 555 universal equilibrium 546 velocity scale 555 Kronecker delta 705 Kronecker tensor 35 Kutta-Joukowsky Transformation 268 Laminar Boundary 624 Blasius equation 625 Faulkner-Skan equation 628 Hartree 628 Polhausen Approximate 629 Laval nozzle 95, 96, 98, 109 Linear wall function 580 Losses due to blade profile diffusion factor 185 effect of Reynolds number 186 integral equation 179, 181 momentum equation , 205, 212, 214 profile loss , 182, , 192, 217 total pressure loss 176, 182, 183, 190, 194, 197, 198, 200, 201, 203, 205, 223, 227 Losses due to exit kinetic energy, effect of degree of reaction exit flow angle 176, 177, 203, , 216, 222 stage load coefficient Losses due to leakage flow in shrouds 204 induced drag 194, , 201 load function losses due to secondary flows 192 mixing loss , Losses due to trailing edge ejection mixing loss , optimum mixing losses 219 Mach number 88 Material acceleration 692 Material derivative 17 Metric coefficients 703 contravariant 705 covariant 705 mixed metric coefficient 705 Mixing length hypothesis 578 Modeling 372 combustion chambers 372 Modeling of inlet, exhaust, and pipe system 357 continuity 358 inlet 357 momentum 358 physical and mathematical 357 pipe 357, 358, 361, 362 total pressure 358, 359 Modeling the compressor dynamic performance active surge prevention 430

38 722 Index module level 2: row-by-row adiabatic calculation 429 module level 3: row-by-row diabatic compression 436 simulation example 427, 431 simulation of compressor surge 432 Modeling turbomachinery components combustion chambers 341, 346, 348 high pressure compressor 342, 351, 354, 355 high pressure turbine 341, 342 low pressure compressor 342, 353, 355 low pressure turbine 342, 348, 354 modular configuration 342, 351, 353, 355, 356 recuperators 346 twin-spool engine 342 Modular configuration 2, 342, 351, 353, 355, 356 control systems 342, 352 shafts 352, 355 valves 342, 352 Momentum balance in stationary frame 32 Natural transition 517 Navier-Stokes 569 equations 547, 548, 552, 561, 569, 570 operator 569 Navier-Stokes equation for compressible fluids 36 Newtonian fluids 35 Nnormal shock 99, , 117 Nonlinear dynamic simulation 323 numerical treatment 339 one-dimensional approximation 331 Predictor-corrector 339 Runge-Kutta 339 thermo-fluid dynamic equations. 328 time dependent equation of continuity 331 time dependent equation of energy 334 time dependent equation of motion 332 Numerical treatment 330, 339 Nusselt number 660 Oblique shock 109, 115, 116, 118, 121 Orthonormal vectors 704 Osculating parabola 553 Over expansion 98 Peclet number 651 effective 652 turbulent 651 Physical components of a Vector 706 Physical components of vector 706 Plenum 343 Pohlhausen 629 profiles 630 slope of the velocity profiles 630 velocity profiles 630 Polytropic load coefficients 452 Power law 647 Prandtl 619 1/7 th power law boundary layer 619 boundary layer approximation 619 boundary layer theory 624 effective Prandtl number 652 experimental observations 619 mixing length 650 mixing length model 653 molecular Prandtl number 652 number 651 turbulent Prandtl number 651 Prandtl mixing length hypothesis 580 Prandtl number 651 effective 652 turbulent 651 Preheater 367 Principle of material objectivity 34

39 Index 723 Radial equilibrium 291, 295, , 309, 316 derivation of equilibrium equation 292 free vortex flow 291, 316, 317 Rayleigh 101, , 108, 109 Recuperator 346, 369 heat Transfer Coefficient 371 recuperators 367 Reynolds transport theorem 25, 27 Richardson energy cascade 547 Rotating frame continuity equation in 52 energy equation in 55 equation of motion in 53 Rotation 700 Rotation tensor 34 Scalar product of 693 Scales of turbulence length 547 of the smallest eddy 547 time 547 time scale 546 time scale of eddies 548 Shear viscosity 36 Shock tube, dynamic behavior 359 mass flow transients 361, 364 pressure transients 361, 362 shock tube dynamic behavior 361 temperature transients Shocks 98, 101, 113, 115, 117 detached shock 117 Hugoniot relation normal shock wave 99, 109, 115 oblique shock wave 115 Prandtl-Meyer expansion 118, 121 strong shock 112, 114, 115, 117 weak shock 114, 117 Similarity condition 627 Spatial differential 691 Spectral tensor 558 Stage parameters 129, 138, 140 circumferential velocity ratio 139, 141 degree of reaction 130, 131, , 137, 139, 140 flow coefficient 129, 137, 139, 140, 142 load coefficient 129, 130, meridional velocity ratio 139, 141 Stage-by-stage and row-by-row adiabatic compression 409 calculation of compression process 409 generation of steady state performance map 418 off-design efficiency calculation 415 rotating stall , row-by-row adiabatic compression 409, 411, 423 surge 385, , surge limit 420, 422, 423, 428, 434 Stanton number 660 Streamline curvature method 291, 292, 300, 301, 306, 309, 310, 312, 313 application of 292, 300, 310 derivation of equilibrium equation 292 examples 306 forced vortex flow 317 free vortex flow 291, 316 special cases 292, 316 step-by-step solution procedure 302 Stress tensor 34 apparent 547 Substantial change 690 Taylor 553 eddies 551 frozen turbulence 551 hypothesis 551 micro length scale 553 time scale 553 Temporal change 690 Tensor 568, 685 contraction 686, 689