TR-CFD-13-77 Variable Normalization (nondimensionalization and scaling) for Navier-Stokes equations: a practical guide Marc Montagnac Technical Report V. 1.0 October 013 Copyrighted by the author(s) Centre Européen de Recherche et de Formation Avancée en Calcul Scientifique 4 avenue Coriolis 31057 Toulouse Cedex 1 France Tél : +33 5 61 19 31 31 Fax : +33 5 61 19 30 00 http://www.cerfacs.fr e-mail: marc.montagnac@cerfacs.fr
Contents 1 Introduction 3 Notation 3 3 System of reference variables 4 3.1 Physics-based argument............................... 4 3. Algebraic argument.................................. 6 3.3 Density-Pressure-Temperature (DPT) scheme................... 7 3.4 Pressure-Temperature-Velocity (PTV) scheme................... 7 3.5 Density-Temperature-Velocity (DTV) scheme................... 7 4 Examples of practical applications 7 4.1 DPT scheme with (ρ, p, T ).............................. 8 4. PTV scheme with (p, T, U)............................. 9 4.3 DTV scheme with (ρ, T, U)............................. 10 4.4 PTV scheme with (p, T, c).............................. 11 4.5 DTV scheme with (ρ, T, c).............................. 1 4.6 DTV scheme with (ρ i, T i, c i )............................. 13 5 Conclusions 14 Page of 14
1 Introduction The Navier-Stokes equations are dimensionally homogeneous. Then, any arbitrary coherent system of units can be used to perform the numerical resolution of these equations. The main system is the Système International d Unités, or SI Units. If the system of units is not required by the software used to solve the equations, then choosing the system of units is the responsibility of the users who must ensure consistency between the different variables involved. This paper focuses on the unit-transformation that gives dimensionless and normalized variables from a given physical reference state. Notation The dimension of a physical quantity X, noted [X], can be expressed as a product of the basic physical dimensions raised to a rational power, [M i L j T k Θ l ]. The main dimension used hereafter and recommended by the SI standard are mass, length, time, and absolute temperature, represented by symbols M, L, T, and Θ. Table 1 gives some examples of physical quantity dimensions. A physical quantity can be expressed in many units, but its dimension will always Table 1: Constant and characteristics of air (standard atmosphere) Quantity Dimension SI r [L T Θ 1 ] 87.053 J.kg 1.K 1 C s [Θ] 110.4 K ρ ML 3 1.50 kg.m 3 p [ML 1 T ] 1.013 10 +5 P a T [Θ] 88.15 K µ [ML 1 T 1 ] 1.78938 10 5 kg.m 1.s 1 remain the same regardless of the unit chosen to measure it. So, any coherent system of units can be used for physical variables and the passage from one system to the other is called a unit-transformation. The unit-transformation is dimensionless otherwise changing the unit system would change the dimensions. As said in the introduction, this paper presents various techniques to normalize some physical variables involved in the Navier-Stokes equations. This involves to introduce three types of variables, namely the dimensionalized variables expressed preferably in SI units and noted without subscript and superscript, for example t, the reference variables chosen for the normalization and noted with the subscript ref, for example t ref, and finally the normalized variables that are by definition with no units and noted with the superscript, for example t. Page 3 of 14
Throughout the paper, the set of Navier-Stokes is closed with the equation of state for a calorically perfect gas with a constant specific heat ratio γ = C p /C v = 1.4. The state equation is given by p = ρrt with the static pressure p, the static temperature T and the specific gas constant r = (γ 1)C v. The internal energy e = C v T only depends on the temperature. The dynamic viscosity µ is given by the Sutherland s formula: µ = µ s ( T T s ) 3 T s + C s T + C s = µ s T 1 + C s /T s T s 1 + C s /T (1) where µ s is the dynamic viscosity at the reference temperature T s and C s is a constant. Other useful variables and numbers can be found in Table : Table : Definition of variables and numbers sound velocity: c = γrt total energy: e = 1/(γ 1) p/ρ + 1 U = rt/(γ 1) + 1 U specific enthalpy: stagnation pressure: h = e + p/ρ = C v T + p/ρ p i = p(1 + 1 (γ 1)M ) γ γ 1 stagnation temperature: T i = T (1 + 1 (γ 1)M ) stagnation density: ρ i = ρ(1 + 1 (γ 1)M ) 1/(γ 1) stagnation specific enthalpy: h i = C v T i + p i /ρ i = γc v T i = γ γ 1 p/ρ + 1 U sound velocity at stagnation conditions: c i = c(1 + 1 (γ 1)M ) 1 Mach number: Reynolds number: M = U/c Re = ρu L/µ 3 System of reference variables The physical quantities must be normalized to manage only dimensionless variables and so a set of reference variables needs to be chosen. Two approaches to the problem are addressed in the following. The first one is developed with an argumentation based on physics whereas the second one is more algebraic. But in fact both methods are closely related. 3.1 Physics-based argument Physical laws are invariant with respect to the system of units used to measure the physical variables. The equation describing mathematically a physical law must also be invariant to scaling in variables within a given system of units. Page 4 of 14
The non-dimensionalized state equation must have the same characteristics as the physical state equation then p = ρrt p/p ref = ρ/ρ ref r/r ref T/T ref which implies that the dimensions of the reference quantities involved must comply with Eq. (). [p ref ] = [ρ ref ] [r ref ] [T ref ] () Similarly, the non-dimensionalized energy equation must have the same characteristics as the physical energy equation then E = r γ 1 T + 1 U E/E ref = r/r ref γ 1 T/T ref + 1 (U/U ref ) which implies that the dimensions of the reference quantities involved must comply with Eqs. (3) and (4). [E ref ] = [r ref ][T ref ] (3) [E ref ] = [U ref ] (4) Combining Eqs. (3) and (4) gives [r ref ][T ref ] = [U ref ]. (5) To sum up, the system composed of Eqs. () and (5) contains 5 dimensions. Then 3 dimensions must be given. It should be underlined that we have arbitrarily chosen to focus on the five following variables (ρ ref, p ref, T ref, U ref, r ref ) and the corresponding dimensions. The dimension [r ref ] is omitted since it is not natural to handle it (but it is really a matter of choice). 3 dimensions still remains to be chosen from 4 that is C4 3 = 4 possibilities. Imposing the choice ([p ref ], [ρ ref ], [U ref ]) is clearly not allowed by Eqs () and (5), and finally 3 types of normalization methods are obtained based on the three following choices: ([p ref ], [ρ ref ], [T ref ]) named Density-Pressure-Temperature (DPT) normalization method, ([p ref ], [T ref ], [U ref ]) named Pressure-Temperature-Velocity (PTV) normalization method, and ([ρ ref ], [T ref ], [U ref ]) named Density-Temperature-Velocity (DTV) normalization method. The reference length L ref is added for all methods. Thus, the physical variables can be normalized by imposing four reference variables that have the four required dimensions. Page 5 of 14
3. Algebraic argument The second approach is based on the Buckingham π theorem. Roughly speaking, the theorem states that if the equation of a physical law involves n physical variables, and the rank of the matrix of dimensions is k, then the original equation is equivalent to an equation involving a set of p = n k dimensionless parameters constructed from the original variables. In other words, the equation gives p dimensionless parameters and only k other variables have to be given to build a nondimensionalization scheme. Let us come back to the present problem. Table 3 gives the dimensional matrix D coming from the five following variables (ρ ref, p ref, T ref, U ref, r ref ) that have been previously selected. This matrix is composed of the values of the exponents of the dimensions involved in each variable, see Table 1. So k = Rank(D) and therefore p = dim Null(D), the dimension of the Table 3: Dimensional matrix ρ ref U ref p ref T ref r ref M 1 0 1 0 0 L -3 1-1 0 T 0-1 - 0 - Θ 0 0 0 1-1 null space of D. Thus, the kernel of the dimensional matrix, Ker(D), has to be found. It describes p ways in which the column vectors of matrix D can be combined to produce a zero vector. [ ] The row augmented matrix D is first constructed where I is the n n identity matrix. I [ ] Then, the column echelon form of that augmented matrix, CEF (D) = B C, is computed. A basis of the null space of D consists in the non-zero columns of C such that the corresponding column of B is a zero column. In the example of Table 3, 1 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 CEF (D) = 1 0 0 0 1 0 1 0 0 0 0 0 0 1 0 0 1 1 1 0 0 0 1 1 Then the null space is generated by three vectors: ker(d) = {0, Page 6 of 14 0 0 1 1, 1 0 1 1 1 }.
The first non null vector corresponds to a variable with dimension ρ 0 U p 0 T 1 r 1 and the second one to a variable with dimension ρ 1 U 0 p 1 T 1 r 1. This gives two dimensionless parameters from the given variables. The first vector is [r][t ] [U] that is equivalent to Eq. (5) and [p] the second vector is that is equivalent to Eq. (). [ρ][r][t ] Note that the column echelon form of the augmented matrix is not unique and other sets of dimensionless parameters, probably less physically significant, could have been generated instead. 3.3 Density-Pressure-Temperature (DPT) scheme If variables with dimensions ([p ref ], [ρ ref ], [T ref ]) are given, then Eq. () gives [r ref ] = [p ref ] and then Eq. (5) gives [U ref ] = [ρ ref ]. 3.4 Pressure-Temperature-Velocity (PTV) scheme [p ref ] [ρ ref ][T ref ] If variables with dimensions ([p ref ], [T ref ], [U ref ]) are given, then Eq. (5) gives [r ref ] = [U ref ] and then Eq. () gives [ρ ref ] = [p ref ] [U ref ]. 3.5 Density-Temperature-Velocity (DTV) scheme [T ref ] If variables with dimensions ([ρ ref ], [T ref ], [U ref ]) are given, then Eq. (5) gives [r ref ] = [U ref ] and then Eq. () gives [p ref ] = [ρ ref ][U ref ]. 4 Examples of practical applications [T ref ] As shown in Sec 3, some reference variables have to be given whatever the normalization method used. Some examples can be found in the following sections. Of course, there are many other possibilities. Page 7 of 14
4.1 DPT scheme with (ρ, p, T ) Let us give the static pressure p, the static temperature T, and the density ρ. So following Sec. 3.3, the reference variables are chosen as p ref ρ ref T ref r ref U ref p p ρ T p /ρ = r ref T ρ T So now the non-dimensionalized (normalized) free stream variables can be set as follows p = 1 ρ = 1 T = 1 r = 1 U = M γ c = γ C v = 1/(γ 1) ( ρ ρ ref ) ( r r ref ( M c U ref ) = p /(ρ T ) r ref ) ( r γ 1 = Cv C s /T ref r ref ) C s = C s /T T s = T s /T µ = M γl /Re Re = ρ U L /µ E = 1/(γ 1) + 1 γm p i = (1 + 1 (γ 1)M ) T i = 1 + 1 (γ 1)M γ γ 1 If the Mach number M and the Reynolds number Re are known at the free stream condition, which are typical flight-type data available in external aerodynamic, then the far-field boundary condition can be set with the previous normalized variables. Page 8 of 14
4. PTV scheme with (p, T, U) Let us give the static pressure p, the static temperature T, and the velocity U. So following Sec. 3.4, the reference variables are chosen as p ref T ref U ref r ref ρ ref p T U U T p U So now the non-dimensionalized free stream variables can be set as follows p = 1 T = 1 U = 1 ρ = γm ( r = 1 γm C 1 v = γ(γ 1)M C s = C s /T T s = T s /T µ = γm L /Re E 1 = γ(γ 1)M + 1 p i = (1 + 1 (γ 1)M ) T i = 1 + 1 (γ 1)M γ γ 1 γp c ρ ref ) ( r r ref ( r γ 1 = Cv r ref ) C s T ref Page 9 of 14
4.3 DTV scheme with (ρ, T, U) Let us give the static density ρ, the static temperature T, and the velocity U. So following Sec. 3.5, the reference variables are chosen as ρ ref T ref U ref p ref r ref ρ T U ρ U U T So now the non-dimensionalized free stream variables can be set as follows ρ = 1 T = 1 U = 1 p = 1 γm r = 1 γm C 1 v = γ(γ 1)M C s = C s /T T s = T s /T µ = 1/Re E = p i = 1 γm 1 γ(γ 1)M + 1 (1 + 1 (γ 1)M ) γ γ 1 ρ c γ p ref p ρ T r ref r γ 1 = Cv r ref C s T ref T i = 1 + 1 (γ 1)M Page 10 of 14
4.4 PTV scheme with (p, T, c) Let us give the static pressure p, the static temperature T, and the sound velocity c. This reference system is a variation of the system presented in Sec 4. in which the reference velocity is the sound velocity. Recall that the quantity U ref must have the dimension LT 1. So the reference variables are chosen as p ref T ref U ref r ref ρ ref c p p T c T c So now the non-dimensionalized free stream variables can be set as follows p = 1 T = 1 c = 1 U = M ρ = γ r = 1 γ C 1 v = γ(γ 1) C s = C s /T T s = T s /T µ = γm L /Re E 1 = γ(γ 1) + 1 M p i = (1 + 1 (γ 1)M ) T i = 1 + 1 (γ 1)M γ γ 1 Page 11 of 14
4.5 DTV scheme with (ρ, T, c) Let us give the static density ρ, the static temperature T, and the sound velocity c. This reference system is a variation of the system presented in Sec 4.3 in which the reference velocity is the sound velocity. Recall that the quantity U ref must have the dimension LT 1. So the reference variables are chosen as ρ ref T ref U ref p ref r ref ρ T c ρ c c T So now the non-dimensionalized free stream variables can be set as follows ρ = 1 T = 1 c = 1 U = M p = 1 γ r = 1 γ C 1 v = γ(γ 1) C s = C s /T T s = T s /T µ = M L /Re E 1 = γ(γ 1) + 1 M p i = 1 γ (1 + 1 (γ 1)M ) T i = 1 + 1 (γ 1)M γ γ 1 Page 1 of 14
4.6 DTV scheme with (ρ i, T i, c i ) Let us give the stagnation density ρ i, the stagnation temperature T i, and the stagnation sound velocity c i. This reference system is a variation of the system presented in Sec 4.5 in which the static quantities are replaced by the stagnation quantities. Recall that the quantities ρ ref, T ref, and U ref must have respectively the dimensions ML 3, Θ, and LT 1. So the reference variables are chosen as ρ ref T ref U ref p ref r ref ρ i T i c i ρ i c i c i T i So now the non-dimensionalized free stream variables can be set as follows ρ = (1 + 1 (γ 1)M ) 1/(γ 1) T = (1 + 1 (γ 1)M ) 1 c = (1 + 1 (γ 1)M ) 1 U = M c p = 1 γ (1 + 1 (γ 1)M ) γ γ 1 r = 1 γ C 1 v = γ(γ 1) C s = C s /T i T s = T s /T i µ = M (1 + 1 (γ 1)M ) 1/(γ 1) 1 L /Re E 1 = ( γ(γ 1) + 1 M )(1 + 1 (γ 1)M ) 1 p i = 1 γ T i = 1 Page 13 of 14
5 Conclusions This paper has presented and explained how to normalize physical quantities in the context of the Navier-Stokes equations. These equations are dimensionally homogeneous and any arbitrary coherent system of units are convenient to solve these equations. Six unit-transformations depending on a given physical reference state have been expressed as examples to give dimensionless and normalized variables, but other ones can be derived on purpose. Page 14 of 14