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<p>In this paper, we establish the existence of a weak solution for the steady RANS model, where the Navier-Stokes equations are coupled with the equation governing the turbulent kinetic energy (TKE). This coupling of the two equations occurs both through the eddy viscosity and eddy diffusion coefficients and also through a Dirichlet boundary condition on a portion of the boundary leading to energy production. Additionally, the right-hand side in the TKE equation contains a term <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(\nu (k) |\nabla \varvec{u}|^2\)</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mrow> <mi>ν</mi> <mrow> <mo stretchy="false">(</mo> <mi>k</mi> <mo stretchy="false">)</mo> </mrow> <mo stretchy="false">|</mo> <mi mathvariant="normal">∇</mi> <mrow> <mi mathvariant="bold-italic">u</mi> </mrow> <mo stretchy="false">|</mo> </mrow> <mn>2</mn> </msup> </math></EquationSource> </InlineEquation> that is only integrable. The proof of the existence of a weak solution is obtained by regularization of the original problem, relaxing the terms that are not sufficiently regular. Additionally, due to the non-homogeneous Dirichlet boundary condition on a portion of the boundary for the energy variable <i>k</i>, we introduce a change of variable for <i>k</i> using the lifting of its value on the boundary of the domain. This implies a new variational formulation with regularization of terms arising from this lifting.</p>

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Analysis of Coupled Steady Navier-Stokes and Turbulent Kinetic Energy Equations with Velocity-Dependent Boundary Conditions

  • 期刊论文

  • 发表日期：

    2026年2月26日

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  • Hamza Boukili,
  • Aziz Takhirov,
  • Driss Yakoubi

摘要

In this paper, we establish the existence of a weak solution for the steady RANS model, where the Navier-Stokes equations are coupled with the equation governing the turbulent kinetic energy (TKE). This coupling of the two equations occurs both through the eddy viscosity and eddy diffusion coefficients and also through a Dirichlet boundary condition on a portion of the boundary leading to energy production. Additionally, the right-hand side in the TKE equation contains a term \(\nu (k) |\nabla \varvec{u}|^2\) ν ( k ) | ∇ u | 2 that is only integrable. The proof of the existence of a weak solution is obtained by regularization of the original problem, relaxing the terms that are not sufficiently regular. Additionally, due to the non-homogeneous Dirichlet boundary condition on a portion of the boundary for the energy variable k, we introduce a change of variable for k using the lifting of its value on the boundary of the domain. This implies a new variational formulation with regularization of terms arising from this lifting.