2. 34 R. Rannacher and V. Heuveline
1.1 The Wankel Motor
Unlike the conventional combustion engines with reciprocating piston motion,
the Wankel Rotary Combustion Engine (RCE) is based on the continuous
rotation of the so-called rotor [BEN,AB]. It has four phases in its combustion
cycle: intake, compression, power and exhaust.
Fig. 1. Four combustion phases: intake, compression, power and exhaust
The fuel air mixture is swept along, so the four phases take place in differ-
ent areas of the engine. Intake and exhaust timing is accomplished directly
by the motion of the rotor which avoids the moving parts of the classical
combustion engines such as valves, cams and timing belts. The most specific
feature in the design of the Wankel motor is probably the peripheral envelope,
an epitrochoid curve which is generated by rolling a circle around another
one. This geometry allows to constantly keep the three rotor tips in contact
with the envelope during the eccentric rotation of the rotor and therefore to
separate the three chambers. The rotor is designed to minimize the surface
to volume ratio in the combustion chamber in order to increase the thermo-
dynamical efficiency. This is best achieved by assuming the biggest possible
rotor contour in the peripheral envelope. Unfortunatly such a contour would
separate the combustion chamber in two parts during the power phase. A
so-called pocket is therefore moulded in the rotor as shown on Fig. 2. In our
numerical simulations we assumed the pocket profile to be parabolic. The
specifications of the Wankel RCE in our numerical simulations as proposed
by the industrial partner (Wankel Rotary GmbH) are given in Table 1.
Generating radius 8.65 cm
Eccentricity 1,35 cm
Chamber width 6,5 cm
Compression ratio 1:7.4
Basic engine speed 3600 rpm
Table 1. Specifications of the Wankel RCE Fig. 2. Pocket locations
3. Flow Simulation in the Wankel Motor 35
The maximum rotation velocity of this prototype is approximatly Vmax ~
30 m/s which corresponds to a flow behavior in the low-Mach-number range
as Mamax < 0.1 in this case. Accordingly acoustic waves may be neglected
which is achieved by the "low-Mach-number" approximation of the compress-
ible Navier-Stokes equations.
1.2 The Mathematical Model
The flow in the rotary engine is described by the compressible Navier-Stokes
equations in the "low-Mach-number" approximation. Here, the pressure p
is split into a thermodynamic part Pth which is constant in space and a
hydrodynamic part Phyd, while only the latter one is used in the equation
of state (see e.g. [MAJ]). Due to the temporal variation of the domain, the
thermodynamic pressure is time dependent, too. We denote by Dt := Ot+v,V'
the material derivative. The whole system of equations written in primitive
variables takes the following form:
T- 1
DtT - V'·v
pDtV - V'. (J.UT) + V'Phyd
pCpDtT - V' {XV'T) - OtPhyd - J..l!7: V'v
= pthlatpth
Pfe
= OtPth
(1)
(2)
(3)
where !7 := (V'v+ V'vT
) - ~(V' ·v)l, cp, >., fe, and J..l represent the shear-
stress tensor, the heat capacity, the heat conductivity, the volume forces and
the dynamic shear viscosity, respectively. The equation of state is given by
(4)
The time derivative of Pth is obtained by first averaging the continuity
equation (1) in space and then substituting Dt T by using (4), whereas the
heating due to Phyd and J..l!7: V'v is neglected. This leads to the following
scalar ODE for Pth :
where Pth(O) = Po is a given initial value. Our numerical approach to sys-
tem (1)-(3) is based on its "variational" formulation which will be briefly
described below. Let, (.,.) denote the usual L2-scalar product on [l. fur-
ther, Hl([l) is the space of L2-functions with generalized (in the sense of
distributions) first-order derivatives in L2([l). This is the only notation from
mathematical theory of function spaces we are going to use in this paper.
The variational formulation of the stationary form of the system (1)-
(3) is obtained by multiplying the equations by appropriate test functions
{X, ,¢, 7f} =: ¢ and integrating over the domain [l. This leads us to define
4. 36 R. Rannacher and V. Heuveline
the stationary semi-linear form a(·;·) by
a(u;¢) := (T-1v·VT,X) - (V·v,X)
+ (p(v·V)v,'lj;) + (j.LVV, V'lj;) - (Phyd - ~j.LV·v, V·'lj;) - (P!e,'lj;)
+ (pcpv·VT,1r) + ()..VT, V1r) - (j.L(j:VV,1r).
In the diffusive terms, we have used integration by parts. Neumann-type
boundary conditions are implicitly represented by the variational formulation,
while Dirichlet boundary conditions have to be explicitly imposed on the
solution. The variational form of system (1)-(3) then reads in short terms:
Find u(t) := {T(t), V(t),Phyd(t)} E V+Ub, such that u(O) = Uo and
(QOtU, ¢) +a(u; ¢) = F(¢) V¢ E V. (6)
where Q is a suitable coefficient matrix multiplying the time derivatives. The
right-hand side F(·) contains the slave variable Pth given by the relation
(5), while P is determined through the modified gas law (4). The term
Ub represents prescribed boundary data. The function space V in which
U - Ub is sought is the tensor product of certain subspaces of H 1
([]) for
temperature and velocity while the space for the hydrodynamic pressure is
L2([]) . If Dirichlet conditions for the velocity are imposed along the entire
boundary, the hydro-dynamic pressure is only defined modulo a constant and
the corresponding pressure-space is L2
([])/ R.
2 The Solution Approach
2.1 The Galerkin Finite Element Discretization
Our Navier-Stokes solver uses a fully implicit approach for solving the "low-
Mach-number" approximation of the compressible Navier-Stokes system (6)
(for a detailed description we refer to [BBR], [BR]).
The Galerkin finite element method is defined on quadrilateral/hexahedral
meshes 7 h = {K} covering the domain t'l. The trial and test spaces Vh C V
consist of continuous, piecewise polynomial vector functions (so-called Qp-
elements) for all unknowns,
Vh = {{T,v,p} E C(t'l)1+dH IT,vIK E Q~+d,pIK E Qs},
where s = 1 for r = 2, and s = r - 2 for r ~ 3. Here, Qr is the space
of (isoparametric) tensor-product polynomials of degree r (for a detailed de-
scription of this standard construction process see [BS]). In order to facilitate
local mesh refinement and coarsening, we allow the cells in the refinement
zone to have nodes which lie on faces of neighboring cells (Fig. 3). The degrees
of freedom corresponding to such "hanging nodes" are eliminated from the
system by interpolation inforcing global conformity (Le., continuity across
interelement boundaries) for the finite element functions. For simplicity, we
do not allow varying polynomial order accross hanging nodes.
5. Flow Simulation in the Wankel Motor 37
K
Fig. 3. Quadrilateral mesh patch with a "hanging node"
The computation of the strongly anisotropic flow in the rotary engine
requires a high resolution in the hole domain which is very time consuming
on varying meshes. This difficulty is accounted for by the use of higher-
order finite element trial functions with varying orders ("hp-method"). The
resulting algebraic systems are lower dimensional with densly filled matrices
such that in some stages "direct" algebraic solvers can be used.
We note that by chosing the trial functions for the pressure of (suffi-
ciently) lower degree than of those for the velocity the form a(·;·) is stable
on the discrete spaces Vh (uniformly in h), i.e., it satisfies the uniform
"Babuska-Brezzi inf-sup-stability" condition. This particularly guarantees a
stable approximation of the pressure. In the case of equal-order trial functions
for v and p, e.g., the popular Ql/Q1-ansatz, the scheme requires "pressure
stabilization" . In addition, the dominant convection is stabilized by the usual
SUPG approach ("streamline upwinding Galerkin", [HB]). Following this
idea, we introduce additional least-squares terms in the continuity equation
(least-squares stabilization and streamline diffusion) and in all other equilib-
rium equations.
In order to formulate this approach in short terms, we write the original
system (1)-(3) in the compact form L(u)u = f(u) with a nonlinear operator
L(·). Then, the stabilization process comprises in the modification of the
original semi-linear form a(uh; </J) by a mesh-dependent semi-linear form:
a6(uh; </J) := a(uh; </J) + (LUh, S</J)6,
with a differential operator S which can be chosen in different ways. Here,
we use S = -L*, and take 8K '" hK proportional to the local mesh size,
1: _ (11QK
1
11 min{JL,A} IVloo,K)-l
UK - a k + h'k + hK
'
with a constant a '" 0.5, and k denoting a time step when the equations
are truely non-stationary or time-stepping is used in solving the stationary
equations. The 8-dependent inner product is defined as usual by
(U,V)6:= L 8K(u,V)K.
KE/h
Accordingly, in the stationary case, we seek a Uh E Vh +Vb,h , satisfying
6. 38 R. Rannacher and V. Heuveline
Clearly, this ansatz is "consistent" in the sense that the additional terms
vanish at the exact solution. This modification serves several purposes: it
stabilizes the pressure in the low-Mach-number approximation, it stabilizes
the possibly strong convection in the flow, and finally it enhances local mass
conservation. This leads to a stable scheme for a wide range of flow conditions.
2.2 Solution of the Algebraic Systems
The nonlinear algebraic system (7) is solved by the damped Newton method.
Denoting the derivative of a(·;·) taken at a discrete function Uh E Vh by
a'(uh;', '), the arising linear systems have the form
Here, w~ is the correction and r~ the equation residual of the preceding
approximation u~. The updates u~+l := u~ + w~ are carried until con-
vergence. The linear subproblems (8) are solved by the GMRES method
with preconditioning by a multigrid iteration. This multigrid component uses
blockwise Gauss-Seidel or ILU smoothing in which the physical unknowns
are implicitly coupled on the cell level. For a description of the details of this
approach, we refer to [BR]. In computing really nonstationary flows this tide
coupling may be lifted by "operator splitting" due to the better conditioning
in this case.
2.3 Tests of Solver Components
The discretization described above has first been tested for various model
problems of interior flows including the usual "lid-driven cavity" for Re =
2000 and a new heat-transfer benchmark "tempertature-driven cavity" for
Ra = 106
comprising large temperature gradients (Fig. 4).
v = l6x'(1 -x)' adiabatic / v =0
V(O,y) = 0 v(l,y) =0
TIwt =960K
v=o
T""d ~ 240K
v=o
0.5
U====ad:zi.=b.='i=C/=V<::==O===:::zI
x
Fig. 4. Configuration of the two flow benchmarks: "lid-driven cavity" (left),
and "temperature-driven cavity" (right)
7. Flow Simulation in the Wankel Motor 39
Discretization by higher-order finite elements. Table 2 shows some
representative results for the simple test case "lid-driven cavity" , while those
for the harder test case "heat-driven cavity" are presented in Table 3. It
turns out that higher-order finite elements have good potential for accurately
computing interior flows even in the presence of strong layers. The solution of
the algebraic problems is the bottleneck in using higher-order finite elements
within implicit flow solvers. This problem is tackled by using hierarchical
multilevel techniques with blockwise smoothers. However, further develop-
ment is necessary at this point to reach fully satisfactory solution efficiency.
I FE ansatz I Fr(Uh) I #cells I #dofs I #entries I CPU I
Q2/Q1 -0.08657 16,384 14,8739 5,994,249 2,100 s
Q3/Q1 -0.08657 4,096 78,723 4,537,737 1,620 s
Q4/Q2 -0.08657 1,024 37,507 3,237,897 3,240 s
Q6/Q4 -0.08657 256 23,043 3,701,769 4,620 s
Q8/Q6 -0.08658 64 10,851 2,779,785 5,400 s
Table 2. Efficiency of higher-order finite elements for solving the "lid-driven cavity"
problem with Re = 2000 (error rv 1%)
CPU I
Q2/Q1 8.843 16,384 214,788 12,069,136 12,324 s
Q3/Q1 8.859 4,096 115,972 9,558,544 8,765 s
Q4/Q2 8.871 1,024 54,148 6,587,904 7,484 s
Q5/Q3 8.893 256 22,084 3,707,536 6,843 s
I FE ansatz I N(Uh) I #cells I #dofs I # entries I
Table 3. Accuracy of higher-order finite elements for solving the "temperature-
driven cavity" problem with Ra = 106
(error rv 1%)
~,I
:.~~
......
·.-........·~
·...............·....
Fig. 5. Convection flow in the "heat-driven-cavity": velocity norm (left) and
temperature (right)
8. 40 R. Rannacher and V. Heuveline
The time stepping. The simulation of the fully nonstationary flow behavior
in the rotary engine, at first, requires a geometric description of the chamber
movement. This is obtained in terms of a transformation <P~' between the
time-dependent domains,
Instead of working on a fixed domain, we consider a variational formulation
and discretization on the deforming domain, nt, i.e. the L2
-scalar products
in (6) are considered on !tt (see [LTD. The discretization is applied in two
steps: A spatial semi-discretization of (6) leads to a set of ordinary differential
equations which is solved by using a stiffiy-stable time stepping scheme like,
for example the implicit Euler, the damped Crank-Nicholson, or the second-
order Fractional-Step-O scheme. Compared to the usual formulation on a
fixed domain here an additional term has to be added in the variational
formulation (6) which represents the time-variation of the domain nt . After
spatial semi-discretization this results in the ODE system
for all <Ph E Vh, with the so-called "mesh-velocity operator" N(t). This
operator corresponds to the matrix
where IJtt := <I>;l , with Jacobian DlJtt , and {4>di=l,...,1 is a basis of the finite
element space on the mesh for nt,. In order to cope with (time dependent)
local singularities arising at the intake and exhaust, we use local mesh refine-
ment in these areas (Fig. 7). The mesh refinement is driven by local "error
indicators" derived heuristically from properties of the computed solution.
3D-solver components. As a preparatory step for the extension of our
Navier-Stokes solver to 3D geometries, several program components for 3D-
mesh handling have been implemented:
Fig. 6. Regular 3D meshes for the Wankel motor (half a rotation cycle)
9. Flow Simulation in the Wankel Motor 41
- a mesh generator for the Wankel motor configuration (see Fig. 6),
- administration of locally refined meshes,
- assembling of system matrices,
- mesh transfer operations for multigrid and mesh numbering.
3 Application to the Wankel Motor Configuration
The experimental 2D code has been used for simulating a quasi-stationary
model of the Wankel motor, Le., for a series of different geometries the flow
is driven by a prescribed inflow at a small inlet and is left free at an outlet.
Inlet
(Dirichlet)
/..... I
no lip boundary E~hausl
( eu ann)
--
Fig. 7. Velocity norm of the flow in the Wankel motor for quasi-stationary
flow conditions and locally refined meshes (half of a rotation cycle)
10. 42 R. Rannacher and V. Heuveline
4 Conclusion and Outlook
We have described the main components of a new numerical tool for simu-
lating gas flow and heat transfer in a rotary engine. The method is based
on the low-Mach-number approximation of the compressible Navier-Stokes
equations and uses a least-squares stabilized finite element discretization of
variable order. For the sake of robustness the approach is fully implicit and
largly exploits multi-level techniques. At first, a "stationary" flow solver has
been developed for interior flows in simple geometries. The natural convection
in a box under strong temperature gradients is a prototypical test case. Then,
similar computations have been performed for the (still stationary) "Wankel
motor" in 2D. The next step is the integration of this quasi-stationary solver
component into a time-stepping scheme for computing really nonstationary
flows following the variation of the domain. Subsequently, a first 3D version
of the code will be compiled.
References
[AB] J. Abraham, D. Ramoth, J. Mannisto, 3-D steady-state wall heat flux and
thermal analysis of a stratified-charge rotary engine, SAE, Society of Automative
Engineers, Technical Paper Series, Nr. 910706, 1991.
[BEN] W.-D. Bensinger, Rotationskolben-Verbrennungsmotoren, Springer-Verlag,
1973.
[BBR] R. Becker, M. Braack, R. Rannacher: Numerical simulation of laminar
flames at low Mach number by adaptive finite elements, Combust. Theory Mod-
elling. 3, 503-534, 1999.
[BR] M. Braack, R. Rannacher, Adaptive finite element methods for low-Mach-
number flows with chemical reactions, Lecture Series 1999-03, 30th Computa-
tional Fluid Dynamics, The von Karman Institute, Belgium, 1999.
[BS] S. C. Brenner and R. L. Scott (1994), The Mathematical Theory of Finite
Element Methods, Springer, Berlin-Heidelberg-New York.
[HB] T. J. R. Hughes and A. N. Brooks, Streamline upwind/Petrov-Galerkin for-
mulations for convection dominated flows with particular emphasis on the in-
compressible Navier-Stokes equation, Comput. Meth. Appl. Mech. Engrg., 32,
pp. 199-259, 1982.
[HRS] P. Houston, R. Rannacher, E. Siili: A posteriori error analysis for stabilised
finite element approximations of transport problems, Preprint 99-26, SFB 359,
University of Heidelberg, to appear in J. Comput. Mech. (1999).
[LT] P. Lesaint and R. Touzani: Approximation of the heat equation in a variable
domain with application to the Stefan problem, SIAM J. Numer. Anal. 26, 366-
379 (1989).
[MAJ] A. Majda, Compressible fluid flow and systems of conservation laws in sev-
eral space varables Springer-Verlag, New York, 1984.
[Rl] R. Rannacher: Adaptive finite element methods for flow problems, Proc. 88th
Int. Symp. on Comput. Fluid Dynamics, Bremen, Sept. 5-10, 1999, to appear in
J. Japan Society of CFD.
[R2] R. Becker, M. Braack, R. Rannacher: Adaptive finite element methods for
flow problems, Proc. Foundations of Computational Mathematics 99, Cambridge
University Press 2000
![34 R. Rannacher and V. Heuveline
1.1 The Wankel Motor
Unlike the conventional combustion engines with reciprocating piston motion,
the Wankel Rotary Combustion Engine (RCE) is based on the continuous
rotation of the so-called rotor [BEN,AB]. It has four phases in its combustion
cycle: intake, compression, power and exhaust.
Fig. 1. Four combustion phases: intake, compression, power and exhaust
The fuel air mixture is swept along, so the four phases take place in differ-
ent areas of the engine. Intake and exhaust timing is accomplished directly
by the motion of the rotor which avoids the moving parts of the classical
combustion engines such as valves, cams and timing belts. The most specific
feature in the design of the Wankel motor is probably the peripheral envelope,
an epitrochoid curve which is generated by rolling a circle around another
one. This geometry allows to constantly keep the three rotor tips in contact
with the envelope during the eccentric rotation of the rotor and therefore to
separate the three chambers. The rotor is designed to minimize the surface
to volume ratio in the combustion chamber in order to increase the thermo-
dynamical efficiency. This is best achieved by assuming the biggest possible
rotor contour in the peripheral envelope. Unfortunatly such a contour would
separate the combustion chamber in two parts during the power phase. A
so-called pocket is therefore moulded in the rotor as shown on Fig. 2. In our
numerical simulations we assumed the pocket profile to be parabolic. The
specifications of the Wankel RCE in our numerical simulations as proposed
by the industrial partner (Wankel Rotary GmbH) are given in Table 1.
Generating radius 8.65 cm
Eccentricity 1,35 cm
Chamber width 6,5 cm
Compression ratio 1:7.4
Basic engine speed 3600 rpm
Table 1. Specifications of the Wankel RCE Fig. 2. Pocket locations](data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7)