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Cristopher Morales Ubal¹, Jeroen van Oijen¹, Nijso Beishuizen^1,2

¹Eindhoven University of Technology

²Bosch Thermotechnology

Department of Mechanical Engineering, section Power & Flow

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Motivation

Extending the preconditioning approach for low-Mach Navier-Stokes equations for multicomponent flows in SU2.

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stream 2: T=300K

stream 1: T=300K

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Energy equation for reacting flows

Extending the preconditioning approach for low-Mach Navier-Stokes equations for multicomponent flows in SU2.

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[1] T. Poinsot, D. Veynante, Theoretical and Numerical Combustion, Edwards, 2005. URL https://books.google.nl/books?id=cqFDkeVABYoC

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Energy equation for reacting flows

Where the sensible enthalpy is given by:

And the specific heat capacity at a constant pressure of the gas mixture is given by:

Term 1: Variations of Pressure
Term 2: Viscous heating
Term 3: Volumetric heat sources
Term 4: Body forces

Extending the preconditioning approach for low-Mach Navier-Stokes equations for multicomponent flows in SU2.

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Energy equation for reacting flows: low-Mach Approximation

Where the sensible enthalpy is given by:

And the specific heat capacity at a constant pressure of the gas mixture is given by:

Pressure term vanishes.
Viscous heating is assumed negligible.
No Volumetric heat sources.
No Body forces.

Extending the preconditioning approach for low-Mach Navier-Stokes equations for multicomponent flows in SU2.

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Energy equation for reacting flows: low-Mach Approximation

Extending the preconditioning approach for low-Mach Navier-Stokes equations for multicomponent flows in SU2.

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Energy equation for reacting flows: low-Mach Approximation

Using (6):

We can write equation (4) as an equation for temperature.

Extending the preconditioning approach for low-Mach Navier-Stokes equations for multicomponent flows in SU2.

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Energy equation for reacting flows: Temperature

Where the heat release due to chemical reactions is given by:

Extending the preconditioning approach for low-Mach Navier-Stokes equations for multicomponent flows in SU2.

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Energy equation for reacting flows: Temperature

Where the heat release due to chemical reactions is given by:

If we assume for now that:

No chemical reactions
Diffusion effects are not important

Extending the preconditioning approach for low-Mach Navier-Stokes equations for multicomponent flows in SU2.

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Energy equation for non-reacting flows: Temperature

Extending the preconditioning approach for low-Mach Navier-Stokes equations for multicomponent flows in SU2.

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Energy equation for non-reacting flows: Temperature

Currently, in SU2, we solve the following energy equation for incompressible flows:

However, we would like to solve (8) instead of (9) for multicomponent flows.

Extending the preconditioning approach for low-Mach Navier-Stokes equations for multicomponent flows in SU2.

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Energy equation for non-reacting flows: Temperature

Currently, in SU2, we solve the following energy equation for incompressible flows:

However, we would like to solve (8) instead of (9) for multicomponent flows.

Then, to do it, we divide (8) by and use derivative rules.

Extending the preconditioning approach for low-Mach Navier-Stokes equations for multicomponent flows in SU2.

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Energy equation for Non-reacting flows: Temperature

which is the energy equation that we are going to use in the preconditioning approach.

Extending the preconditioning approach for low-Mach Navier-Stokes equations for multicomponent flows in SU2.

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Preconditioning: Conservative form

Following the steps done by Economon [2], we write the N-S equations as follows:

where the conservative variables are:

and the convective, viscous and source terms are:

Extending the preconditioning approach for low-Mach Navier-Stokes equations for multicomponent flows in SU2.

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[2] T.D.Economon, Simulation and Adjoint-based Design for Variable Density Incompressible Flows with Heat Transfer,

Multidisciplinary Analysis and Optimization Conference, 2018. https://arc.aiaa.org/doi/abs/10.2514/6.2018-3111

Instead of

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Preconditioning: Non-Conservative form

Then, we transform the N-S equations from Conservative to Working variables:

where the working variables are:

and:

Extending the preconditioning approach for low-Mach Navier-Stokes equations for multicomponent flows in SU2.

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Preconditioning: Preconditioning matrix

Where:

Extending the preconditioning approach for low-Mach Navier-Stokes equations for multicomponent flows in SU2.

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Preconditioning: Preconditioning matrix

Extending the preconditioning approach for low-Mach Navier-Stokes equations for multicomponent flows in SU2.

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Preconditioning: Preconditioning matrix

Preconditioning matrix:

Extending the preconditioning approach for low-Mach Navier-Stokes equations for multicomponent flows in SU2.

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Preconditioning: Preconditioning matrix

where:

Extending the preconditioning approach for low-Mach Navier-Stokes equations for multicomponent flows in SU2.

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Test case: 2D Incompressible mixing channel system

Extending the preconditioning approach for low-Mach Navier-Stokes equations for multicomponent flows in SU2.

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Non-slip adiabatic walls

Symmetry plane

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First simulation: Two streams with equal temperature

Extending the preconditioning approach for low-Mach Navier-Stokes equations for multicomponent flows in SU2.

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Non-slip adiabatic walls

Symmetry plane

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First simulation: Two streams with equal temperature

Extending the preconditioning approach for low-Mach Navier-Stokes equations for multicomponent flows in SU2.

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Develop

Extension

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Second simulation: Two streams with different temperatures

Extending the preconditioning approach for low-Mach Navier-Stokes equations for multicomponent flows in SU2.

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Non-slip adiabatic walls

Symmetry plane

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Second simulation: Two streams with different temperatures

Extending the preconditioning approach for low-Mach Navier-Stokes equations for multicomponent flows in SU2.

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Develop

Extension

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Third simulation: Switching temperatures of the streams

Extending the preconditioning approach for low-Mach Navier-Stokes equations for multicomponent flows in SU2.

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Non-slip adiabatic walls

Symmetry plane

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Third simulation: Switching temperatures of the streams

Extending the preconditioning approach for low-Mach Navier-Stokes equations for multicomponent flows in SU2.

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Develop

Extension

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Residuals temperature: Third simulation

Extending the preconditioning approach for low-Mach Navier-Stokes equations for multicomponent flows in SU2.

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Conclusions

Extending the preconditioning approach for low-Mach Navier-Stokes equations for multicomponent flows in SU2.

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Preconditioning for multicomponent flows is extended to account for variable heat capacity.
Implementation in SU2 is verified and validated.

Outlook:

To be done: pull request in SU2 + adding diffusivity term.
Next steps: couple CANTERA to SU2 and implement detailed chemistry for incompressible flows.

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Extending the preconditioning approach for low-Mach Navier-Stokes equations for multicomponent flows in SU2.

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Questions?

Develop

Extension

stream 2: T=300K

stream 1: T=300K

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Energy equation for reacting flows: low-Mach Approximation

Using (6):

The unsteady and convective terms in (4) can be written as follows:

Extending the preconditioning approach for low-Mach Navier-Stokes equations for multicomponent flows in SU2.

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Energy equation for reacting flows: low-Mach Approximation

Extending the preconditioning approach for low-Mach Navier-Stokes equations for multicomponent flows in SU2.

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Energy equation for reacting flows: low-Mach Approximation

Extending the preconditioning approach for low-Mach Navier-Stokes equations for multicomponent flows in SU2.

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Energy equation for reacting flows: Temperature

Where the heat release due to combustion is given by:

Extending the preconditioning approach for low-Mach Navier-Stokes equations for multicomponent flows in SU2.

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Energy equation for reacting flows: Temperature

Where the heat release due to combustion is given by:

If we assume for now that:

No combustion.
Diffusion effects are not important.

Extending the preconditioning approach for low-Mach Navier-Stokes equations for multicomponent flows in SU2.

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Energy equation for Non-reacting flows: Temperature

Extending the preconditioning approach for low-Mach Navier-Stokes equations for multicomponent flows in SU2.

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Energy equation for Non-reacting flows: Temperature

Currently, in SU2, we solve the following energy equation for incompressible flows:

However, we would like to solve (12) instead of (13) for multicomponent flows.

Extending the preconditioning approach for low-Mach Navier-Stokes equations for multicomponent flows in SU2.

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Energy equation for Non-reacting flows: Temperature

Currently, in SU2, we solve the following energy equation for incompressible flows:

However, we would like to solve (12) instead of (13) for multicomponent flows.

Then, to do it, we divide (12) by .

Extending the preconditioning approach for low-Mach Navier-Stokes equations for multicomponent flows in SU2.

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Energy equation for Non-reacting flows: Temperature

Extending the preconditioning approach for low-Mach Navier-Stokes equations for multicomponent flows in SU2.

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Energy equation for Non-reacting flows: Temperature

Using derivation rules, the right-hand size can be written as:

The energy equation (14) for multicomponent flows becomes:

Extending the preconditioning approach for low-Mach Navier-Stokes equations for multicomponent flows in SU2.

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Preconditioning: Projected convective flux

The projected convective flux jacobian is:

Extending the preconditioning approach for low-Mach Navier-Stokes equations for multicomponent flows in SU2.

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Preconditioning: Preconditioned convective flux jacobian

The preconditioned convective flux jacobian is:

Extending the preconditioning approach for low-Mach Navier-Stokes equations for multicomponent flows in SU2.

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Preconditioning: Eigenvalues

The eigenvalues of the preconditioning system:

Which are the same eigenvalues obtained by Economon [2].

Extending the preconditioning approach for low-Mach Navier-Stokes equations for multicomponent flows in SU2.

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