Dispersed multiphase flows are dynamic systems of two or more phases, at least one of which is dispersed in a fluid in the form of fine particles. Systems of this kind are omnipresent in nature and technology. Weather patterns, ocean currents and blood circulation are just a few examples of their numerous occurrences in nature, while their importance in engineering is illustrated by examples such as drive systems, energy and heat engineering and manufacturing technologies. The numerical modelling of such systems is particularly challenging in terms of the mathematical representation of the disperse phase and the associated physical phenomena such as turbulence and various mechanisms of mass, momentum and energy exchange.

QBMM are based on the macroscopic description in the form of moments of the NDF. The moment equations can be derived directly from the underlying PBE. Taking into account only one internal coordinate \(\xi\) as well as advection and diffusion with the drift function \(\mathcal{A}(\xi)\) and diffusivity \(\mathcal{D}(\xi)\), the NDF \(n\) over time \(t\) follows the equation

\(\frac{\partial n(\xi, t)}{\partial t} = - \frac{\partial }{\partial \xi} \left[ \mathcal{A}(\xi) \, n(\xi, t) \right] \; + \; \frac{\partial^2 }{\partial \xi^2} \left[ \mathcal{D}(\xi) \, n(\xi, t) \right]\).

If the definition of the \(k\)-th moment \(m_k = \int_\Omega \xi^k n(\xi) \mathrm{d} \xi\), the following results for the \(k\)-th moment equation

\(\frac{\mathrm{d} m_k(t)}{\mathrm{d} t} = - k \int_\Omega \xi^{k-1} \, \mathcal{A}(\xi) \, n(\xi, t) \mathrm{d} \xi \; + \; k(k-1) \int_\Omega \xi^{k-2} \, \mathcal{D}(\xi) \, n(\xi, t) \mathrm{d} \xi\).

The moment equations cannot be solved in this form without knowledge of the NDF. QBMM offer a solution to this closure problem by applying quadrature formulas, i.e.

\(\frac{\mathrm{d} m_k(t)}{\mathrm{d} t} \approx - k \sum\limits_j w_j \, \xi_j^{k-1} \, \mathcal{A}(\xi_j) \; + \; k(k-1) \sum\limits_j w_j \, \xi_j^{k-2} \, \mathcal{D}(\xi_j)\).

If the quadrature nodes \(\xi_j\) and weights \(w_j\) are the nodes and weights of the Gaussian quadrature with the weight function \(n\), the method for closing the moment equations is referred to as the quadrature method of moments (QMOM) [1]. This is the basis of numerous derived quadrature methods, see [2]. All these methods have in common that the quadrature formula sought can be calculated exclusively from the known information in the form of moments. As part of our research, we have developed an extension of the QMOM by applying the anti-Gauss quadrature formulas presented by Laurie [3], which in particular should improve the quality of the approximation for strongly nonlinear problems.

The general basis for the numerical solution of the problems discussed here are the reactive Euler equations for several species. These are in their three-dimensional conservative form:
\begin{gather}
\frac{\partial \mathbf{U}}{\partial t}
+ \frac{\partial \mathbf{F}(\mathbf{U})}{\partial x}
+ \frac{\partial \mathbf{G}(\mathbf{U})}{\partial y}
+ \frac{\partial \mathbf{H}(\mathbf{U})}{\partial z}
= \mathbf{S}(\mathbf{U}) \,, \label{eq:eulerEquation} \\
\mathbf{U} = \begin{bmatrix}
\rho \\ \rho u \\ \rho v \\ \rho w \\ \rho E \\ \rho Y_s
\end{bmatrix} ,\,
\mathbf{F} = \begin{bmatrix}
\rho u \\ \rho u^2+p \\ \rho uv \\ \rho uw \\ u(\rho E+p) \\ \rho u Y_s
\end{bmatrix} ,\,
\mathbf{G} = \begin{bmatrix}
\rho v \\ \rho uv \\ \rho v^2+p \\ \rho vw \\ v(\rho E+p) \\ \rho v Y_s
\end{bmatrix} ,\,
\mathbf{H} = \begin{bmatrix}
\rho w\\ \rho uw \\ \rho vw \\ \rho w^2+p \\ w(\rho E+p) \\ \rho w Y_s
\end{bmatrix} ,\, \label{eq:eulerComponents}
\end{gather}

where \(\rho\) is the density, \(\mathbf{u}=(u,v,w)\) the velocity, \(E=e+||\mathbf{u}||^2/2\) the total energy, \(e\) the internal energy, \(p\) the pressure and \(Y_s,\,s=1,\ldots,N_s-1\) the mass fractions of the reactive species, which are normalised so that they sum to one. The above equations are closed by the approximation as a calorically perfect gas \(p=\rho(\gamma-1)e\), where \(\gamma=c_p/c_v\) is the ratio of the specific heats. The term \(\mathbf{S}(\mathbf{U})\) is a placeholder for any possible source term that takes into account, for example, external forces, chemical reactions or any supply of mass, momentum, energy or species. To compute the numerical solution of the above equations, we use fully conservative finite volume schemes discretised on block-structured Cartesian grids. Thanks to the use of the external C++ library AMReX [5] in our CFD solver, we can locally refine the grid in the regions of interest using AMR techniques [1]. To ensure the stability of small or irregular cells resulting from the embedding of complex geometries in the structured grids, we use a conservative cut-cell method [3].

Aircraft engines must be sustainably improved in terms of emissions, efficiency and noise. This research project aims to combine the enormous potential of artificial intelligence and machine learning with the latest methods of optimisation and process integration in order to design the design assistance systems of the future using engine technology as an example.

External partners:

• Rolls-Royce Germany
• TU Berlin
• TH Brandenburg
• FRIENDSHIP SYSTEMS

Chairs involved at the BTU:

• Chair of Numerical Mathematics and Scientific Computing (Head)
• Chair of Structural Mechanics and Vehicle Vibrational Technology
• Chair of Aeroengine Design
• Chair of Applied Mathematics

• Chair of Polymer-based Lightweight Design