Adjoint method for transport of heat and species in internal automotive flows
Paper in proceeding, 2015

The continuous adjoint formulation of the steady-state Navier-Stokes equations with scalar transport has been implemented in the open-source framework OpenFOAM™. The flow is solved using a steady-state compressible flow solver. A cost function based on uniform species distribution at a selected region on the surface of the geometry is presented. The continuous adjoint approach is applied to calculate the gradient of the cost function with respect to normal motion of the surface geometry. The main advantage of applying the adjoint method is the considerable decrease in simulation time compared to traditional methods, reducing the total simulation time for the whole sensitivity field to roughly two flow simulations. The current implementation of the adjoint equations with scalar transport is applied to a simple two dimensional channel. The sensitivity of a species distribution at a location at the surface of the channel with respect to the normal motion of the surface is evaluated. The results from the adjoint implementation are validated against gradients approximated using finite difference calculations. The comparison shows close agreement between the gradients obtained using the two methods, implying that the implementation of the adjoint equations can be used as a guideline when improving a design or in an optimization process for this type of flow.

Sensitivity analysis

Continuous adjoint method

Scalar transport

Computational fluid dynamics



Chalmers, Applied Mechanics, Fluid Dynamics

Sinisa Krajnovic

Chalmers, Applied Mechanics, Fluid Dynamics

Proceedings of the Thermal and Fluids Engineering Summer Conference

23791748 (eISSN)

Vol. 2015-August 299-309
9781567004311 (ISBN)

1st Thermal and Fluid Engineering Summer Conference, TFESC 2015
New York City, USA,

Subject Categories

Applied Mechanics

Computational Mathematics

Fluid Mechanics and Acoustics

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