Application of the PITM Method Using Inlet Synthetic Turbulence Generation for the Simulation of the Turbulent Flow in a Small Axisymmetric Contraction

Abstract

We investigate the turbulence modeling of second moment closure used both in RANS and PITM methodologies from a fundamental point of view and its capacity to predict the flow in a low turbulence wind tunnel of small axisymmetric contraction designed by Uberoi and Wallis. This flow presents a complex phenomenon in physics of fluid turbulence. The anisotropy ratio of the turbulent stresses τ ₁₁/τ ₂₂ initially close to 1.4 returns to unity through the contraction, but surprisingly, this ratio gradually increases to its pre-contraction value in the uniform section downstream the contraction. This point constitutes the interesting paradox of the Uberoi and Wallis experiment. We perform numerical simulations of the turbulent flow in this wind tunnel using both a Reynolds stress model developed in RANS modeling and a subfilter scale stress model derived from the partially integrated transport modeling method. With the aim of reproducing the experimental grid turbulence resulting from the effects of the square-mesh biplane grid on the uniform wind tunnel stream, we develop a new analytical spectral method of generation of pseudo-random velocity fields in a cubic box. These velocity fields are then introduced in the channel using a matching numerical technique. Both RANS and PITM simulations are performed on several meshes to study the effects of the contraction on the mean velocity and turbulence. As a result, it is found that the RANS computation using the Reynolds stress model fails to reproduce the increase of anisotropy in the centerline of the channel after passing the contraction. In the contrary, the PITM simulation predicts fairly well this turbulent flow according to the experimental data, and especially, the “return to anisotropy” in the straight section of the channel downstream the contraction. This work shows that the PITM method used in conjunction with an analytical synthetic turbulence generation as inflow is well suited for simulating this flow, while allowing a drastic reduction of the computational resources.

Appendices

Appendix A: Computational Resources for DNS

It is worth evaluating, as a rough guide, the necessary computer resources for simulating the flow in the experimental wind tunnel of Uberoi and Wallis [3] at the Reynolds number R e ≈ 4.47 × 10⁵ in terms of number of grid-points and computational times. DNS simulations require that the grid-size is at least of order of magnitude of the Kolmogorov scale η computed as η = (ν ³/𝜖)^1/4. We consider here a simple model where the turbulent kinetic and the dissipation-rate evolve along the centerline of the channel with the coordinate x ₁ according to the power law decay of homogeneous turbulence leading to the equations

$$ k(x_{1}) = k_{0} \left[ 1+ \frac{\epsilon_{0}} {n k_{0} U_{b}} x_{1} \right]^{-n} $$

(30)

and

$$ \epsilon(x_{1}) = \epsilon_{0} \left[ 1+ \frac{\epsilon_{0}} {n k_{0} U_{b}} x_{1} \right]^{-(n+1)} $$

(31)

where $n= 1/(c_{{\epsilon }_{2}}-1)$ and k(x ₀) = k ₀, 𝜖(x ₀) = 𝜖 ₀. In the case of an infinitesimal domain of the channel between x ₁ and x ₁ + d x ₁, the number of grid-points d N ₁(x ₁)d N ₂ d N ₃ of the mesh is then given by

$$ dN_{1}(x_{1}) dN_{2} dN_{3} =\frac{64\, {D^{2}_{1}} dx_{1}} {\eta^{3}(x_{1})} $$

(32)

in order to describe a “minimal” sine curve on a full period using at least four grid points. The number of grid-points is then computed by integration of Eq. 32. As a result, it is found that

$$ N_{1} N_{2} N_{3}= N_{2} N_{3} {\int}_{0}^{L_{1}} dN_{1} dx_{1} = N_{2} N_{3} \frac{64\, {D^{2}_{1}}} {{\eta^{3}_{0}}} {\int}_{0}^{L_{1}} \left[ 1+ \frac{\epsilon_{0}} {n k_{0} U_{b}} x_{1} \right]^{-\frac{3(n+1)} {4}} dx_{1} $$

(33)

Hence,

$$ N_{1} N_{2} N_{3} = \frac{64\, {D^{2}_{1}}} {{\eta^{3}_{0}}} \frac{ k_{0} U_{b}} {\epsilon_{0}} F \left( \frac{ \epsilon_{0} L_{1}} {n k_{0} U_{b}} \right) $$

(34)

where

$$ F = \frac{4n} {1-3n} \left[\left( 1+ \frac{\epsilon_{0}} {n k_{0} U_{b}} L_{1} \right)^{\frac{1-3n} {4}} -1 \right] $$

(35)

The computational time is proportional to the number of grid points N ₁ N ₂ N ₃ and the number of temporal iterations N _it and the time required by the central processing unit t _CPU per iteration and per grid-point, leading to the result t = N ₁ N ₂ N ₃ N _{i t} t _CPU. The number of iteration is given by N _it = T/δ t where T is the convective time allowing the eddies to move towards the exit of the channel, whereas δ t is given by the CFL condition δ t = η ₀/U _b that is here a more stringent constraint than the Kolmogorov time-scale $\tau _{0}=\sqrt {\nu /\epsilon _{0}}={\eta ^{2}_{0}}/\nu$ by a factor about 100. The convective time is computed from T = L ₁/U _b so that the computational time is therefore given by

$$ t = N_{1} N_{2} N_{3} \frac{L_{1}} {\eta_{0}}\, t_{\text{CPU}} $$

(36)

The turbulent Reynolds number is $R_{t}=k^{2}/\nu \epsilon = L_{e}\sqrt {k}/\nu =(L_{e}/\eta )^{4/3}$. Using the preceding values U _b = 11.0 m/s, k ₀ = 0.46 m²/s and 𝜖 ₀ = 10 m ²/s ³ the turbulence length-scale and the Kolmogorov scale are computed by $L_{e}=\nu R_{t}/\sqrt {k}$ so that in the entrance of the channel, L e ₀ ≈ 3.43 cm and η ₀ ≈ 0.135 mm, respectively. From Eqs. 34 and 35, it is found that the number of grid points is order of N _η ≈ 6 × 10¹², and that the dimensionless computational time t/t _CPU ≈ 10¹⁷. These numerical order of magnitudes clearly show that DNS and also by extension highly resolved LES are not at all affordable in term of computational resources even with the rapid increase of super computer power.

Appendix B: Generation of Anisotropic Turbulence with an Imposed Energy Spectrum

1.1 B.1 Generation of isotropic turbulence in the spectral space

The first step consists in generating an homogeneous isotropic field in a cubic box of dimension L using the method developed by Roy [49]. In this aim, we define a random vector stream function in the spectral space [61, 62]

$$ \hat{\boldsymbol{\psi}}(\boldsymbol{\kappa}) = a(\boldsymbol{\kappa}) \boldsymbol{\hat{\psi}_{1}} (\boldsymbol{\kappa}) +j b(\boldsymbol{\kappa}) \boldsymbol{\hat{\psi}_{2}}(\boldsymbol{\kappa}) $$

(37)

where $\boldsymbol {\hat {\psi }_{1}} (\boldsymbol {\kappa })$ and $\boldsymbol {\hat {\psi }_{2}} (\boldsymbol {\kappa })$ are two real vectors uniformly distributed on the sphere (see Fig. 22) of radius unity in the half space κ ₃ ≥ 0, a and b are two functions defined as a(κ) = α ₁(κ) cos(λ κ) and b(κ) = α ₂(κ) sin(λ κ) , α ₁ and α ₂ are real functions and λ is a random number in the interval [0, 2π]. The two real random fields $\boldsymbol {\hat {\psi }_{1}} (\boldsymbol {\kappa })$ and $\boldsymbol {\hat {\psi }_{2}} (\boldsymbol {\kappa })$ obtained by drawing lots in the Fourier space define a complex vector stream function $\boldsymbol {\hat {\psi }} (\boldsymbol {\kappa })$ in the spectral space. If working in spherical coordinates, the random vector Ψ ^{n, p} on the sphere of radius unity is obtained by $\Psi ^{n,p}_{1}= \sin \theta _{n} \cos \phi _{p}$, $\Psi ^{n,p}_{2}= \sin \theta _{n} \sin \phi _{p}$ and $\Psi ^{n,p}_{3}= \cos \theta _{n}$, where θ _n and ϕ _p are the polar and azimuthal angles, respectively. The random angles are then given by ϕ _p = 2π p and θ _n = arccos(1 − 2n) where p and n are uniform random numbers in the interval [0,1]. The stream function is then computed in the half space κ ₃ < 0 by the relation $\hat {\boldsymbol {\psi }}(-\boldsymbol \kappa ) = \hat {\boldsymbol {\psi }}^{*}(\boldsymbol \kappa )$, where $\hat {\boldsymbol {\psi }}^{*}$ denotes the conjugate of $\hat {\boldsymbol {\psi }}$, to ensure that the velocity is real. The velocity is therefore obtained by the relation

$$ \hat{\boldsymbol{u}}(\boldsymbol{\kappa}) = \boldsymbol{\kappa} \wedge \hat{\boldsymbol{\psi}} (\boldsymbol{\kappa}) = \alpha_{1} (\boldsymbol{\kappa}) \cos(\lambda \boldsymbol{\kappa}) (\boldsymbol{\kappa} \wedge \boldsymbol{\hat{\psi}_{1}} (\boldsymbol{\kappa}) + j \alpha_{2} (\boldsymbol{\kappa}) \sin(\lambda \boldsymbol{\kappa}) (\boldsymbol{\kappa} \wedge \boldsymbol{\hat{\psi}_{2}} (\boldsymbol{\kappa}) $$

(38)

verifying automatically the continuity equation. In order to satisfy the energy equation,

$$ k= \frac{1} {2} \sum\limits_{\boldsymbol{\kappa}} \,\, \left<\hat{u}_{m}(\boldsymbol{\kappa}) \hat{u}^{*}_{m}(\boldsymbol{\kappa})\right> = \sum\limits_{\kappa} E(\kappa) \delta \kappa $$

(39)

one finds that the spectral velocity correlation tensor must satisfy the spectral energy equation

$$ \left< \hat{u}_{m}(\boldsymbol \kappa) \hat{u}^{*}_{m}(\boldsymbol\kappa) \right> = \left( \frac{2\pi} {L}\right)^{3} \frac{E(\kappa)} {2 \pi \kappa^{2}} $$

(40)

It is straightforward to see that the two functions a(κ) and b(κ) which appear in Eq. 37 are given by

$$ a(\boldsymbol{\kappa}) = \frac{\cos (\lambda \boldsymbol{\kappa})} {|| \boldsymbol{\kappa} \wedge \boldsymbol{\hat{\psi}_{1}} (\boldsymbol{\kappa}) ||} \left( \frac{2\pi} {L}\right)^{\frac{3} {2}} \left( \frac{E(\kappa)} {2 \pi \kappa^{2}} \right)^{\frac{1}{2}} $$

(41)

$$ b(\boldsymbol{\kappa}) = \frac{\sin(\lambda \boldsymbol{\kappa})} {|| \boldsymbol{\kappa} \wedge \boldsymbol{\hat{\psi}_{2}} (\boldsymbol{\kappa}) ||} \left( \frac{2\pi} {L}\right)^{\frac{3} {2}} \left( \frac{E(\kappa)} {2 \pi \kappa^{2}} \right)^{\frac{1}{2}} $$

(42)

where ||.|| denotes the norm. In Eq. 37, the energy spectrum density is modeled by equations

$$ E(\kappa) = \chi \kappa^{m} \,\,\,\,for\,\,\,\, \kappa < \kappa_{0} $$

(43)

$$ E(\kappa) = C_{\kappa} \epsilon^{2/3} \kappa^{-5/3} \,\,\,\, for\,\,\,\,\, \kappa \ge \kappa_{0} $$

(44)

where the coefficient χ is given by

$$ \chi = C_{\kappa} \epsilon^{2/3} \kappa^{(5+3m)/3}_{0} $$

(45)

and the matching wave number κ ₀ is determined by the energy condition $k={\int }^{\infty }_{0} E(\kappa ) d\kappa$ leading to the value

$$ \kappa_{0}= C_{\kappa}^{3/2} \frac{\epsilon} {k^{3/2}} \left[ \frac{3m+5} {2(m+1)} \right]^{3/2} $$

(46)

For κ _c ≥ κ ₀ where κ _c is the cutoff wave number, the ratio value between the subgrid scale energy and the total energy is

$$ \frac{k_{sfs}} {k} = \frac{3(m+1)} {3m+5} \left( \frac{\kappa_{c}} {\kappa_{0}} \right)^{-2/3} $$

(47)

Fig. 22

Random vector fields in the spectral space on the sphere of radius unity

1.2 B.2 Generation of anisotropic turbulence in the spectral space

The second step consists in applying a tensor transformation β on the isotropic field produced by the stream function $\hat {\psi } (\boldsymbol {\kappa })$ to generate an anisotropy field. As for the isotropic case, this procedure allows to satisfy the continuity equation. The velocity is then computed by $\hat {\boldsymbol {u}}= \boldsymbol {\kappa } \wedge \boldsymbol {\beta } \hat {\boldsymbol {\psi }} (\boldsymbol {\kappa })$. It is possible to compute the coefficient β _{i j} by means of algebra calculus in the spectral space. The starting point consists in applying another tensor transformation α on the velocity itself leading to $\hat {\boldsymbol {u}}= \boldsymbol {\alpha } (\boldsymbol {\kappa } \wedge \hat {\boldsymbol {\psi }} (\boldsymbol {\kappa }))$. The issue to address is then to determine the tensor β as a function of α. The resulting equation which must be solved finally reads

$$ \hat{\boldsymbol{u}}(\boldsymbol{\kappa)}= \boldsymbol{\alpha} (\boldsymbol{\kappa} \wedge \hat{\boldsymbol{\psi}} (\boldsymbol{\kappa})) = \boldsymbol{\kappa} \wedge \boldsymbol{\beta} \hat{\boldsymbol{\psi}} (\boldsymbol{\kappa}) $$

(48)

or for the i component,

$$ \hat{u}_{i}=j \kappa_{j} \alpha_{im} \epsilon_{mjk} \hat{\psi}_{k} =j \epsilon_{ijk} \kappa_{j} \beta_{km} \hat{\psi}_{m} $$

(49)

Equation 49 allows to work on the velocity as well as on the stream function field. The spectral turbulent energy associated with the wave number κ is then given by

$$ \left< \hat{u}_{i} \hat{u}^{*}_{i} \right> = \alpha_{im} \alpha_{in} \epsilon_{mjk} \epsilon_{npq} \kappa_{j} \kappa_{p} \left< \hat{\psi}_{k} \hat{\psi}^{*}_{q} \right> = \epsilon_{ijk} \epsilon_{ipn} \beta_{km} \beta_{nq} \kappa_{j} \kappa_{p} \left< \hat{\psi}_{m} \hat{\psi}^{*}_{q} \right> $$

(50)

For simplification purposes, we consider now that α and β are diagonal matrices. Moreover, as $\hat {\boldsymbol {\psi }} (\boldsymbol {\kappa })$ is an isotropic vector, the right hand-side of Eq. 50 becomes

$$ \left< \hat{u}_{i} \hat{u}^{*}_{i} \right> = \epsilon_{ijk} \epsilon_{ipn} \beta_{kk} \beta_{kk} \kappa_{j} \kappa_{p} \left< \hat{\psi}_{k} \hat{\psi}^{*}_{k} \right> $$

(51)

or in a developed form,

$$ \left< \hat{u}_{1}\hat{u}^{*}_{1} \right> = \beta^{2}_{33} {\kappa^{2}_{2}} \left< \hat\psi_{3} \hat\psi^{*}_{3}\right>+ \beta^{2}_{22} {\kappa^{2}_{3}} \left< \hat\psi_{2} \hat\psi^{*}_{2} \right> $$

(52)

$$ \left< \hat{u}_{2} \hat{u}^{*}_{2} \right> = \beta^{2}_{11} {\kappa^{2}_{3}} \left< \hat\psi_{1} \hat\psi^{*}_{1}\right> + \beta^{2}_{33} {\kappa^{2}_{1}} \left< \hat\psi_{3} \hat\psi^{*}_{3} \right> $$

(53)

$$ \left< \hat{u}_{3} \hat{u}^{*}_{3} \right> = \beta^{2}_{22} {\kappa^{2}_{1}} \left< \hat\psi_{2} \hat\psi^{*}_{2}\right> + \beta^{2}_{11} {\kappa^{2}_{2}} \left< \hat\psi_{1} \hat\psi^{*}_{1} \right> $$

(54)

To reproduce the axisymmetric turbulence of the Uberoi and Wallis experiment [3] such as 〈u ₂ u ₂〉 = 〈u ₃ u ₃〉, Eqs. 53 and 54 imply that β ₂₂ = β ₃₃. Moreover, to get 〈u ₁ u ₁〉 > 〈u ₂ u ₂〉, these equations lead to β ₁₁/β ₂₂ < 1. Taking into account these conditions, it is found that the matrix β can be written as

$$ \mathbf{\boldsymbol{\beta}}= \left( \begin{array} {cccc} 1-2 \gamma & 0 & 0 \\ 0 & 1+ \gamma & 0 \\ 0 & 0 & 1+ \gamma \end{array} \right) $$

(55)

where γ is a numerical coefficient which must be determined to get the desired ratio anisotropy. Equation 50 allows to compute the coefficient γ and leads to the following equations

$$ \left< \hat{u}_{1} \hat{u}^{*}_{1} \right> = \alpha^{2}_{11} \left[ {\kappa^{2}_{2}} \left< \hat \psi_{3} \hat \psi^{*}_{3}\right> +{\kappa^{2}_{3}} \left< \hat\psi_{2} \hat\psi^{*}_{2}\right> \right] = \beta^{2}_{33} {\kappa^{2}_{2}} \left< \hat\psi_{3} \hat\psi^{*}_{3}\right> + \beta^{2}_{22} {\kappa^{2}_{3}} \left< \hat\psi_{2} \hat\psi^{*}_{2} \right> $$

(56)

$$ \left< \hat{u}_{2} \hat{u}^{*}_{2} \right> = \alpha^{2}_{22} \left[ {\kappa^{2}_{3}} \left< \hat \psi_{1} \hat \psi^{*}_{3}\right> +{\kappa^{2}_{1}} \left< \hat\psi_{3} \hat\psi^{*}_{3}\right> \right] = \beta^{2}_{11} {\kappa^{2}_{3}} \left< \hat\psi_{1} \hat\psi^{*}_{1}\right> + \beta^{2}_{33} {\kappa^{2}_{1}} \left< \hat\psi_{3} \hat\psi^{*}_{3} \right> $$

(57)

$$ \left< \hat{u}_{3} \hat{u}^{*}_{3} \right> = \alpha^{2}_{33} \left[ {\kappa^{2}_{1}} \left< \hat \psi_{2} \hat \psi^{*}_{2}\right> +{\kappa^{2}_{2}} \left< \hat\psi_{1} \hat\psi^{*}_{1}\right> \right] = \beta^{2}_{22} {\kappa^{2}_{1}} \left< \hat\psi_{2} \hat\psi^{*}_{2}\right> + \beta^{2}_{11} {\kappa^{2}_{2}} \left< \hat\psi_{1} \hat\psi^{*}_{1} \right> $$

(58)

Now, with the aim to obtain equations which do not depend anymore on the wave number κ, we define the spherical mean of the Fourier transform $\mathcal {M}$ by the relation [61]

$$ \psi (\kappa) = \mathcal{M}(\psi (\boldsymbol{\kappa})) = \frac{1} {\mathcal{A}} \int \!\!\!\!\ {\int}_{\partial \mathcal{A}} \!\! \psi (\boldsymbol{\kappa}) \,\, d\mathcal{A} $$

(59)

where $\mathcal {A}$ denotes the area element on the sphere of radius κ = |κ|. Applying this operator $\mathcal {M}$ on Eqs. 56, 57 and 58 leads to the resulting equations that read

$$\begin{array}{@{}rcl@{}} && 2 \alpha^{2}_{11} = \beta^{2}_{33}+\beta^{2}_{22} \\ &&2 \alpha^{2}_{22} = \beta^{2}_{11}+\beta^{2}_{33} \\ && 2 \alpha^{2}_{33} = \beta^{2}_{11}+\beta^{2}_{22} \end{array} $$

(60)

As $\boldsymbol {\kappa } \wedge \hat {\boldsymbol {\psi }} (\boldsymbol {\kappa })$ is an isotropic vector whereas $\hat {\boldsymbol {u}}(\boldsymbol {\kappa }) = \boldsymbol {\alpha } \big (\boldsymbol {\kappa } \wedge \hat {\boldsymbol {\psi }}(\boldsymbol {\kappa })\big )$ is an anisotropic vector, the anisotropy of the turbulence in the spectral space can be measured by the ratio

$$ \frac{ \left< \hat{u}_{1}(\kappa) \hat{u}^{*}_{1}(\kappa) \right> } { \left< \hat{u}_{2}(\kappa) \hat{u}^{*}_{2}(\kappa) \right> } = \frac{\alpha^{2}_{11}} {\alpha^{2}_{22}} =\zeta $$

(61)

In this case, for a given value of ζ, the solving of the system (60) gives the results α ₁₁ = 1 + γ ₀, $\alpha _{22}=\alpha _{33}=((5 {\gamma ^{2}_{0}} -2 \gamma _{0} +2)/2)^{1/2}$ and the coefficient γ ₀ is solution of the second degree equation

$$ {\gamma_{0}^{2}}\left( 1-\frac{5} {2} \zeta\right) + \gamma_{0} (2+\zeta)+(1-\zeta) = 0 $$

(62)

Starting with this initial value γ ₀, it is then possible to compute its exact value γ by successive approximations to get the prescribed value 〈u ₁(x)u ₁(x)〉 / 〈u ₂(x)u ₂(x)〉 = ζ.

1.3 B. 3 Return to the physical space

As usually, the velocity in the physical space u(x) is then computed from its inverse Fourier transform given by

$$ u_{i}(x_{1},x_{2},x_{3}) = \sum\limits_{\kappa_{1}=\frac{2 \pi n_{1}} {L}} \sum\limits_{\kappa_{2}=\frac{2 \pi n_{2}} {L}} \sum\limits_{ \kappa_{3}=\frac{2 \pi n_{3}} {L}} \hat{u}_{i}(\kappa_{1},\kappa_{2},\kappa_{3}) \exp{(j \kappa_{m} x_{m}) } $$

(63)

for $(n_{1},n_{2},n_{3}) \in \left [-\frac {N} {2}+1,\frac {N} {2}\right ]$.