Email: matjaz.ramsak@um.si

A short summary of PhD thesis titled

By

*Matjaž Ramšak*

**What is done?**

The doctoral thesis presents a multidomain boundary element method (further 'BEM') for solving a general diffusion convective transport problem using variable diffusivity. Developed BEM solver is applied for turbulence modelling using two equation turbulence model. A turbulent flow over backward facing step is solved which is the first case using BEM. The doctoral thesis has been nominated by CEACM as a finalist in the selection process to the ECOMASS Ph.D. award to the best Ph.D. thesis of 2004 on computational methods in applied sciences and engineering. See diploma and report of selection comittee.

**Why?**

The BEM is relatively young approximation method originated from late seventies. Originally BEM was developed for potential problems governed by Poisson equation. The main advantage of BEM against Finite Element Method ('FEM') and Finite Volume Method ('FVM') is reducing the domain problem to boundary problem using Green's functions. Soon BEM becomes a widely accepted for laminar flows. Practically all engineering fluid dynamics problems are of turbulent nature, but this is the first case where BEM is applied for a complex flow problems using k-epsilon turbulence model.

**How?**

The velocity vorticity and stream vorticity formulation of Navier-Stokes equations are presented and tested. The integral boundary domain equations are discretised using mixed boundary elements and multidomain method also known as subdomain technique [1]. When using continuous elements of high orders and the multidomain method are used, application of the matching conditions at common interfaces leads to a sparse overdetermined system of algebraic equations. These are solved in a least squares sense using fast iterative solver based on method of conjugate gradients [2].

**What we have learned?**

The laminar flow test cases shows higher accuracy of BEM against the other approximation techniques using the same mesh densities [1,3,4]. The accuracy of BEM is more evident on low grid densities. The CPU consumption per iteration became comparable to other numerical methods, while overall CPU is still much higher because of higher number of necessary iterations. The stability of developed BEM is demonstrated using the driven cavity flow at Reynolds number value of 50.000 which is the highest value computed using BEM till to now [3]. Three different k-epsilon turbulent near wall models are tested in the numerical examples [x]. In the turbulent channel flow good agreement is obtained using all three models. The results of turbulent flow over a backward step are in excellent agreement comparing them to FVM and the same turbulent model. We have proved the well-known fact, that the k-epsilon model predictions on recirculation lengths are under predicted when compared to the results of direct numerical simulation.

**Some background and indirect achievements** (for all unquoted references see [3])

Practically all engineering fluid dynamics problems are of turbulent nature. BEM is well known for solving laminar flows but not for turbulent flows. In the open literature one finds BEM numerical model applied to turbulent flows only for the solution of relative simple channel flows using algebraic and one-equation turbulence models, see Škerget et al. 1990, Alujevič et al. 1991. Author Wu and Sugavanam 1978 applied BEM only for flow kinematics while flow kinetics, which is more complicated part in turbulent flows, was solved using finite elements method and low-Reynolds two-equation turbulent model. The solution of simple 2D turbulent Couette flow and flow past a finite plate was presented.

Two-equation turbulent models are most commonly used in engineering practice because of the best ratio between solution quality and computational economy. From the numerical point of view, turbulence modelling using two-equation turbulence model is more complicated than laminar flow modelling. Two additional nonlinear equations are added to the basic nonlinear set of equations governing laminar flow increasing the nonlinearity of basic set of equations across variable turbulent viscosity. In the case of k-epsilon turbulent model two additional equations contains extremely nonlinear source functions connecting them together and slowing down the solution convergence. These source functions became singular near the walls where a special treatment is necessary. From this we can conclude that good numerical algorithm solving laminar flow is necessary and serves as a starting point before the one can proceed to turbulent flows.

The requested properties for laminar flow solver are: (a) stability at higher Reynolds number values, (b) accuracy, (c) economy of the solution and (d) ability to solve high grid density with high ratio longest/shortest element.

The property (a) is a common problem using BEM. The stability of our BEM numerical algorithm is the main topic in our previous work [3] where considerable effort was done to increase the stability at convection dominated laminar flows. The result was a stable algorithm at Reynolds number 50.000 in the case of laminar flow in driven cavity example, thus far exceeding the largest number of 15.000 computed using BEM by Rek and Škerget 1994.

The accuracy of BEM, the property (b), is well known quality of BEM, see Grigoriev and Dargush 1999, Škerget et al. 1999, [1]. In comparison to the other numerical methods the accuracy of BEM is significant at lower grid densities.

All properties described are fulfilled using following techniques. The property (d) is concerned with the available computer memory. If the solution domain is treated as a single entity, which is the primary advantage of BEM against other domain numerical techniques, the system matrix is full and non-symmetric. For diffusive problems the matrix dimension is of order of the number of boundary elements. For diffusion-convection problems, where the domain of solution has to be discretised, the number of domain nodal points significantly increases the matrix dimension. At this stage, already the integrals of fundamental solution are wasteful of memory at higher grid densities. An elegant solution is multidomain BEM, well known also as subdomain technique, see Teles 1987, Hriberšek and Škerget 1996. If the solution domain is divided to the smallest subdomains in the limiting case the obtained system matrix is similar to those obtained using finite element method, namely sparse and diagonal block banded. The indexed matrix storage for nonzero numbers is introduced thus decreasing the necessary memory of order 10.000 times at moderate grid densities, see [2]. At higher grid densities, the saving is even higher. Using the multidomain approach the matrix of fundamental solution integrals are smaller by the same order. Dealing with the maximal number of nodal points fitting inside of 1 GB memory it is increased from 2000 to 500.000 using a multidomain technique computed in [4]. The necessary CPU for one iteration is few minutes on 3.0 GH PC, see [4]. Property (c) considering the economy of computation is also fulfilled using multidomain approach. Sparse system matrix is ideal for fast iterative methods solving algebraic system equations, see Hriberšek and Škerget 1996. The speed-up factor in comparison to direct solvers is approximately 10.000 already at moderate grid densities, [2].

The main restriction of BEM is the availability of Green's functions for partial differential equations with constant coefficients, which is not the case in turbulent flow. The problem is solved using multidomain model. At each subdomain different constant diffusivity is applied. Further, at each subdomain a principle of variable diffusivity is applied dividing diffusivity into the constant and variable part, see Rek and Škerget 1994.

**References**

**1. **RAMŠAK, Matjaž, ŠKERGET, Leopold. Mixed boundary elements for laminar flows. *Int. j. numer. methods fluids*, 1999, vol. 31, pp. 861-877. [COBISS.SI-ID 4884502]

**2. **RAMŠAK, Matjaž, ŠKERGET, Leopold. Iterative least squares methods for solving over-determined matrix for mixed boundary element method.*Zeitschrift für angewandte Mathematik und Mechanik*, 2000, 80, suppl. 3, pp. S657-S658. [COBISS.SI-ID 5306902]

**3. **RAMŠAK, Matjaž, ŠKERGET, Leopold. A subdomain boundary element method for high Reynolds laminar flow using stream function-vorticity formulation. *Int. j. numer. methods fluids*, 2004, vol. 46, pp. 815-847. [COBISS.SI-ID 9214998]

**4. **RAMŠAK, Matjaž, ŠKERGET, Leopold, HRIBERŠEK, Matjaž, ŽUNIÈ, Zoran. A multidomain boundary element method for unsteady laminar flow using stream function-vorticity equations. *Eng. anal. bound. elem.*. [Print ed.], 2005, vol. 29, iss. 1, pp. 1-14.

**5. **RAMŠAK, Matjaž, ŠKERGET, Leopold. A multidomain boundary element method for two equation turbulence models. In process for publication in v*Int. j. numer. methods fluids*