The P-DNS Method for Turbulent Fluid Flows: An Overview

Abstract

An overview of the Pseudo-Direct Numerical Simulation (P-DNS) method is presented. This is a multi-scale method aiming at numerically solving the unknown fields at two different scales, namely coarse and fine. The P-DNS method is built around four key ideas. The first one is that of numerically solving both scales, which facilitates obtaining solutions to problems of both concurrent multi-scale and hierarchical multi-scale types. The second key idea is that of computing off-line the fine solution via Direct Numerical Simulation in simplified domains, termed representative volume elements (RVEs), while the third idea is that of storing the basic (physics-informed) results obtained from this solution in a problem-independent unique dimensionless database. This database may be subsequently used for solving different problems at the coarse level, i.e. by using coarse meshes in the corresponding problem domains, via a surrogate model. In this sense P-DNS resembles Reduced Order Methods, which require a previous off-line evaluation of the modes to be used in the solution, sharing with them the benefit of solving the reduced problem, more precisely the coarse scale, in P-DNS terms, in a very efficient way. The fourth and last key idea of P-DNS is based on the fact that most of the high-frequency modes of a turbulent flow are convected by the fluid velocity of the low-frequency modes. Taking this into account the P-DNS technique is implemented in such a way that the fine instabilities are convected by the velocity field of the coarse solution. Finally, although the P-DNS method has been used to solve different computational mechanics problems, such as convection-diffusion and convection-reaction/absorption problems, the scope of this overview will be limited to its application to turbulent incompressible fluid flows, including both single phase and particle-laden flows.

Appendix 1: Computing the dimensionless parameters and obtaining the equilibrium stress tensor

• 1. Compute the velocity gradient ${\text{G}_{\rm ij}^{\rm org}} = \frac{\partial {u^c_{i}}}{\partial x_j}$ of the coarse velocity field ${\text{u}^{\rm c}_{\rm i}}$.¹

• 2. Obtain the transformation tensor ${S_{ij}}$ and the corresponding rotated velocity gradients $g_{1}$ and $g_{2}$ following the procedures in “The internal RVE case” section for an internal cell or in “The wall-RVE case” section in this appendix for a boundary cell.

• 3. Compute the dimensionless gradients on the rotated configuration as:

  $$\begin{aligned} Id_{1} = g_{1} \frac{\rho H^ 2}{\mu } \\ Id_{2} = g_{2} \frac{\rho H^ 2}{\mu } \end{aligned}$$

• 4. Enter the database with $|Id_{1}|$ and $|Id_{2}|$ (remember that the RVE simulations were computed for positive $Id_i$ values) to obtain the dimensionless equilibrium fine stresses ${\widetilde{T}}_{ij}^{\rho}$.

• 5. Recover the correct sign of the off-diagonal components of ${\widetilde{T}}_{ij}^{\rho}$ by computing

  $$\begin{aligned} {\widetilde{T}}_{xy}^\rho := sign(Id_{1}) {\widetilde{T}}_{xy}^\rho , \\ {\widetilde{T}}_{xz}^\rho := sign(Id_{2}) {\widetilde{T}}_{xz}^\rho , \\ {\widetilde{T}}_{yz}^\rho := sign(Id_{2}){\widetilde{T}}_{yz}^\rho . \end{aligned}$$

• 6. Obtain the dimensional equilibrium stress tensor:

  $$\begin{aligned} {\overline{T}}_{ij}^\rho = {\widetilde{T}}_{ij}^{\rho} \frac{\mu^{2}}{\rho H^ 2}. \end{aligned}$$

• 7. Finally, transform the tensor back to the coarse mesh coordinates:

  $$\begin{aligned} {\overline{T}}_{ij}^\rho = S_{ik} {\overline{T}}_{kl}^\rho S_{jl}. \end{aligned}$$

1.1 The internal RVE case

Starting with a real, symmetric, null trace tensor

$$\begin{aligned} G_{ij}^{symm} = \left( \begin{matrix} g_{11} &{} g_{12} &{} g_{13} \\ g_{12} &{} g_{22} &{} g_{23} \\ g_{13} &{} g_{23} &{} -(g_{11}+g_{22}) \\ \end{matrix} \right) ,\end{aligned}$$

(96)

which is typical of the symmetric part of a velocity gradient, the idea is to obtain, by orthogonal transformations, a tensor

$$\begin{aligned} {G_{ij}^{rot}} = \left( \begin{matrix} 0 &{} g_{1} &{} g_{2} \\ g_{1} &{} 0 &{} g_{2} \\ g_{2} &{} g_{2} &{} 0 \\ \end{matrix} \right) , \end{aligned}$$

(97)

where $|g_{1}|\ge |g_{2}|$. This can be achieved with the following procedure.

• (i) Rotate to the principal axes by computing

  $$\begin{aligned} V_{ik} G_{kl}^{symm} V_{jl} = \text{G}_{\rm ij}^{\sigma } = \left( \begin{matrix} \sigma_{1} &{} 0 &{} 0 \\ 0 &{} \sigma_{2} &{} 0 \\ 0 &{} 0 &{} \sigma_{3} \\ \end{matrix} \right) , \end{aligned}$$

  (98)

  where $\sigma_i$ are the eigenvalues, with $\sigma_{1}\le \sigma_{2} \le \sigma_3$, and where the rotation matrix $V_{ij}$ is built from the corresponding eigenvectors.

• (ii) Reorder the eigenvalues to obtain the configuration of maximum energy, by defining

  $$\begin{aligned} {R^{0}_{ij}} = \left( \begin{matrix} \cos (\theta ) &{} 0 &{} \sin (\theta ) \\ 0 &{} 1 &{} 0 \\ -\sin (\theta ) &{} 0 &{} \cos (\theta ) \\ \end{matrix} \right) , \end{aligned}$$

  where $\theta =90^{\circ}$ if $({\sigma_{1}}{\sigma_{2}})<0$, $\theta =0^{\circ}$ otherwise, and computing

  $$\begin{aligned} {R^{0}_{ik}} G_{kl}^{\sigma } {R^{0}_{jl}} = {\text{G}_{\rm ij}^{0}}. \end{aligned}$$

  (99)

• (iii) Rotate around first axis to make null a diagonal coefficient, by defining

  $$\begin{aligned} R^1_{ij} = \left( \begin{matrix} 1 &{} 0 &{} 0 \\ 0 &{} \cos (\theta ) &{} -\sin (\theta ) \\ 0 &{} \sin (\theta ) &{} \cos (\theta ) \\ \end{matrix} \right) , \end{aligned}$$

  where $\theta =90^{\circ}$ if $|{G_{22}^{0}} |<\varepsilon$, $\theta =\tan^{-1}\left( \sqrt{{G_{11}^{0}}/{G_{22}^{0}} + 1}\right)$ otherwise, and computing

  $$\begin{aligned} R^1_{ik} G_{kl}^{0} R^1_{jl} = G_{ij}^{1}. \end{aligned}$$

  (100)

• (iv) Rotate around third axis to make null the remaining diagonal coefficients, by defining

  $$\begin{aligned} {R^{2}_{ij}} = \left( \begin{matrix} \cos (\theta ) &{} -\sin (\theta ) &{} 0\\ \sin (\theta ) &{} \cos (\theta ) &{} 0\\ 0 &{} 0 &{} 1 \\ \end{matrix} \right) , \end{aligned}$$

  where $\theta^{0}=45^{\circ}$, and computing

  $$\begin{aligned} {R^{2}_{ik}} {G_{kl}^1} {R^{2}_{jl}} = {G_{ij}^{2}} \end{aligned}$$

  (101)

• (v) Rotate to obtain $xy \ne xz$, by defining

  $$\begin{aligned} R^3_{ij} = \left( \begin{matrix} \cos (\theta ) &{} 0 &{} -\sin (\theta )\\ 0 &{} 1 &{} 0 \\ \sin (\theta ) &{} 0 &{} \cos (\theta )\\ \end{matrix} \right) , \end{aligned}$$

  where $\theta^{0} = 90^{\circ}$ if $|{G_{23}}^{2} |\ne |{G_{13}^{2}}|,\;{\theta} = 0^{\circ}$ otherwise, and computing

  $$\begin{aligned} R^3_{ik} G_{kl}^2 R^3_{jl} = G_{ij}^{3}. \end{aligned}$$

  (102)

• (vi) Rotate to obtain $sign(xz) = sign(yz)$, by defining

  $$\begin{aligned} {R^{4}_{ij}} = \left( \begin{matrix} 1 &{} 0 &{} 0 \\ 0 &{} \cos (\theta ) &{} -\sin (\theta ) \\ 0 &{} \sin (\theta ) &{} \cos (\theta ) \\ \end{matrix} \right) , \end{aligned}$$

  where $\theta^{0}=180^{\circ}$ if $|{G_{23}}^{3}-{G_{13}^{3}} |> 0$, $\theta =0^{\circ}$ otherwise, and computing

  $$\begin{aligned} {R^{4}_{ik}} G_{kl}^3 R^4_{jl} = G_{ij}^{4}. \end{aligned}$$

  (103)

Finally, by defining

$$\begin{aligned} {S_{ij}} = R^4_{ik} R^3_{kl} R^2_{lm} R^1_{mn} R^{0}_{np} V_{pj} \end{aligned}$$

the full transformation can be expressed as

$$\begin{aligned} {G_{ij}^{rot}} = S_{ik} G_{kl}^{symm} S_{jl} \end{aligned}$$

(104)

and the opposite transformation as

$$\begin{aligned} G_{ij}^{symm} = S_{ki} G_{kl}^{rot} S_{lj}. \end{aligned}$$

(105)

1.2 The wall-RVE case

In this case, the process starts with a real, null trace but not necessarily symmetric tensor

$$\begin{aligned} {G_{ij}^{org}} = \left( \begin{matrix} g_{11} &{} g_{12} &{} g_{13} \\ g_{21} &{} g_{22} &{} g_{23} \\ g_{33} &{} g_{32} &{} -(g_{11}+g_{22}) \\ \end{matrix} \right) , \end{aligned}$$

(106)

typical of a full velocity gradient, a unit vector $n_{i}$ normal to the wall and a cell-centered velocity $U_i$. The idea is to express ${G_{ij}^{org}}$ in a new local coordinate system, where the unit vectors ${\hat{i}}$, ${\hat{j}}$, ${\hat{k}}$ represent the axes, so that the wall normal is aligned with the ${\hat{j}}$ direction and the tangential velocity ${U_{i}^{\tau}} = U_{i} - (U_{j} n_{j}) n_{i}$ is aligned with the ${\hat{i}}$ direction (Fig. 37).

Fig. 37

Procedure for elements on walls

• 1. Define the auxiliary variables as

  $$\begin{aligned} s= & {} n_{i} {\hat{j}}_i \\ m= & {} {\epsilon_{ijk}} {n_{j}} {\hat{j}}_k \end{aligned}$$

  where ${\epsilon_{ijk}}$ is the Levi–Civita tensor.

• 2. Rotate such as align boundary normal with y-direction

  $$\begin{aligned} R^1_{ij} = s {\delta_{ij}} + (1-s) m_im_j/\sqrt{m_km_k} + n_{i} {\hat{j}}_j - n_j {\hat{j}}_i \end{aligned}$$

  where ${\delta_{ij}}$ is the identity tensor and compute

  $$\begin{aligned} R^1_{ik} G_{kl}^{org} R^1_{jl} = G_{ij}^{1}. \end{aligned}$$

  (107)

• 3. Redefine the auxiliary variables as

  $$\begin{aligned} U^{\prime}_i= & {} R^1_{ij} U_j^\tau \\ s= & {} U^{\prime}_i{\hat{i}}_i \\ m= & {} {\epsilon_{ijk}} U^{\prime}_j{\hat{i}}_k \end{aligned}$$

• 4. Rotate such as align tangential velocity with x-direction

  $$\begin{aligned} R^2_{ij} = s {\delta_{ij}} + (1-s) m_im_j/\sqrt{m_km_k} + U^{\prime}_i {\hat{i}}_j - U^{\prime}_j {\hat{i}}_i \end{aligned}$$

  and compute

  $$\begin{aligned} R^2_{ik} G_{kl}^{1} R^2_{jl} = {G_{ij}^{rot}}. \end{aligned}$$

  (108)

Finally, by defining

$$\begin{aligned} {S_{ij}} = R^2_{ik} R^1_{kj} \end{aligned}$$

the full transformation can be expressed as

$$\begin{aligned} {G_{ij}^{rot}} = S_{ik} G_{kl}^{org} S_{jl} \end{aligned}$$

(109)

It is important to note that:

• The resulting tensor ${G_{ij}^{rot}}$ is aligned with boundary normal and tangential velocities.

• This tensor is expressed as

  $$\begin{aligned} {G_{ij}^{rot}} = \left( \begin{matrix} 0 &{} g_{1} &{} 0 \\ 0 &{} 0 &{} 0 \\ 0 &{} 0 &{} 0 \\ \end{matrix} \right) , \end{aligned}$$

  (110)

  only if the unique non-zero component of ${G_{ij}^{org}}$ is uv (with u aligned to the tangential velocity and v to the wall normal). If this is not the case, the other non-zero components are neglected.