Università di Pisa
Scuola di Doorato “Leonardo da Vinci”
Doctoral Dissertation
inMechanical Engineering
Prediction models for the
dynamical behaviour of
honeycomb gas seals
Diego Saba
Università di Pisa
Scuola di Doorato “Leonardo da Vinci”
A dissertation in partial satisfaction of the requirements for the degree of Doctor of Philosophy
in
Mechanical Engineering
Prediction models for the
dynamical behaviour of
honeycomb gas seals
Author:
Diego Saba . . . .
Advisors:
Prof. Ing. Paola Forte . . . .
Prof. Ing. Enrico Ciulli . . . .
SSD ING–IND/
Abstract
Honeycomb gas seals are used in high performance turbomachinery because their damping capabilities have a favourable stabilizing effect on the rotordy-namics of the machine. e predictability of the dynamic behaviour of the seals is critical at design stage. Currently, an isothermal two-control-volume bulk-flow model is used simulate the dynamical behaviour of honeycomb gas seals. However, the effectiveness of this procedure is not always satisfactory. On the basis of new experimental data from the UHP test rig of GE Oil & Gas in Florence, new predictive models have been devised that take into account the thermal effects in the gas and alternative empirical relations for the friction. A versatile simulation tool has been developed to experiment the effect of new assumptions and speed up the formulation and verification cycle of new em-pirical correlations. As a result of this investigation, a striking insensitivity to all the aempted modifications has been acknowledged. A CFD analysis of the flow was judged useful at this point, to get a beer understanding of the fluid dynamics at the scale of a honeycomb cell. A method was develop to conduct CFD simulations on a single cell, by exploiting the periodic geometry and the quasi-periodic properties of the flow. Fictitious source terms are used in the CFD simulation, based on a first order approximation of the slowly varying part of the flow.
Contents
Introduction
Characterization of honeycomb gas seals
. Dynamic coefficients . . .
. Experimental data . . .
.. Testing apparatus . . .
.. Identification methodology . . .
One-dimensional honeycomb seal models
. Introduction . . .
. One-control-volume bulk flow model . . .
. Two-control-volume bulk flow models . . .
.. Boundary conditions . . .
. Friction model . . .
. Heat transfer model . . .
. Numerical methods . . .
.. Unperturbed equations. . .
.. Exit boundary condition with penalty function . . .
.. Artificial volume viscosity . . .
.. Perturbed equations . . .
. code . . .
. Results . . .
CFD model of a single cell
. Introduction . . .
. Dual-scale analysis . . .
. asi-periodic assumption . . .
. Fictitious volume sources . . .
iv Contents
. Non-stationary simulations . . .
A Formulas used for the CFD simulations
A. Fictitious sources . . .
A. Processing of CFD output . . .
B Nomenclature
C Bibligraphy
C. Papers . . .
C. Books on rotordynamics . . .
C. Books on fluid dynamics . . .
Chapter
Introduction
is dissertation is the crowning effort of a doctoral research scholarship on rotordynamics, funded by GE Oil & Gas, Florence. e research activity was focused on predictive models for the dynamical behaviour of honeycomb gas seals. A new test rig for gas seals was coming on line at the GE plant in Florence, and a substantial amount of experimental data on honeycomb seals was expected in the following years.
Honeycomb seals are used in high performance turbomachinery for their damping capability. Gas seals, in general, tend to have a destabilizing effect on turbomachines, due ultimately to the gas swirling in the seal clearance. Damping, on the contrary, has a stabilizing effect. Damping seals can lead to the design of more compact machines, with equal power and efficiency.
It has long been known that internal seals can have a strong impact on the dynamics of a turbomachine. A destabilizing effect due to the seals was reported for the Kaybob compressor (Smith) and for the Ekofisk com-pressor (Cochrane).
Since the ’s, smooth-rotor/honeycomb-stator annular seals have been used in process centrifugal compressors where they were employed as di-rect replacements for aluminum labyrinth seals that were being consumed by the process fluid. As it turned out, honeycomb seals had significantly less leakage compared to conventional see-through labyrinth seals for the same clearances.
In a smooth-rotor/honeycomb-stator annular seal, a honeycomb paern of hexagonal cavities is present on the stator, Figure.. Hole-paern seals are very similar. e same paern of cavities is present, but the their shape is cylindrical, Figure..
Introduction
Figure .: Honeycomb seal. Figure .: Hole-paern seal.
later. In , honeycomb seals were used to stabilize the High-Pressure Oxygen Turbopump of the Space Shule Main Engine (Childs and Moyer
). Currently, honeycomb seals are oen used for the balance piston seal of a high performance compressor, especially in a back-to-back arrangement. When a seal is positioned half way between the bearings, where the the first mode of vibration of the sha has its maximum amplitude, its dynamic be-haviour becomes of greater importance.
e honeycomb cavities have the effect of an artificially roughened sur-face, and increase friction. us, it is reasonable to dig them on the stator, where they can effectively decrease swirl and stabilize the system.
To predict the dynamic behaviour of honeycomb gas seals, a relatively simple model has been devised by Kleynhans and Childs (Kleynhans ; Kleynhans and Childs), a two-control-volume bulk-flow model that trans-lates to a boundary value problem. However, the validity limits of this model are still under scrutiny. A simulation tool, based on an isothermal ver-sion of this model, was developed at the Turbomachinery Laboratory, Texas A&M University. It was named and is available to GE Oil & Gas.
While waiting for the first experimental results from the test rig, it was planned to develop a new simulation tool based on Kleinhans’ and Childs’ model. Its purpose was to obtain a flexible tool, easy to modify in response to the on-coming experimental feedback. is tool was developed and named . To speed up the debugging process, the new tool was validated, for the isothermal part, against .
e experimental results from the test rig were much more difficult to understand than anticipated. It was expected that some tuning of the em-pirical equations of the bulk-flow model would be sufficient to bring
simula- tions and experiments into agreement. On the contrary, no new idea seemed able to reconcile them.
A beer understanding of the fluid dynamics of the honeycomb cells was desirable, and to this end it was decided to resort to . Instead of trying to solve the dynamic problem for a whole seal, I propose, in this work, to simu-late a single cell of fluid, in a way that can be used to validate new bulk-flow models. Current experiments provide information on the global behaviour of the seal, not on single cells. e CFD of a single cell can fill this gap and support, or disprove, the empiricism involved in a candidate bulk-flow model. Only preliminary results from CFD single-cell simulations are presented. ey show however that the method is feasible.
Chapter
Characterization of honeycomb
gas seals
. Dynamic coefficients
e dynamic behaviour of a rotordynamic component is the way it reacts to the vibrations of the rotor. In most cases the relationship between rotor displacements and reacting forces can be assumed to be linear, and gas seals are no exception. For annular gas seals, the relevant displacements of the rotor are lateral. e reaction to tilting movements is usually neglected and has not yet been the object, as far as I know, of thorough investigations. It is known to play a role in liquid annular seals (Childs).
e linear relationship between displacements and forces is best expressed as a transfer function in frequency domain. It can be wrien as
´ ˆF = H ˆc (.)
where e is the lateral displacement of the rotor, F the force acting on the rotor, and the hat sign over the variables indicates Fourier transform. e transfer function H is a two-by-two matrix whose components are called dynamic coefficients and are functions of the angular frequency ω. Given a fixed reference frame x, y, integral with the stator, the components can be rendered explicit by writing
´ [ˆ Fx ˆ Fy ] = [ Hxx Hxy Hyx Hyy ] [ ˆ cx ˆ cy ] (.) A mechanical system having a spring and a damper in parallel is
de- Characterization of honeycomb gas seals scribed by the equation
´ F = Ke + C ˙c (.)
and, in frequency, by
´ ˆF = (K + iωC)ˆc. (.)
By analogy, stiffness and damping coefficients are defined as
K =ℜH, C = ℑH/ω. (.)
though, in general, they are functions of frequency. Stiffness and damping coefficients are also called dynamic coefficients collectively, so that a spe-cific name for H would be desirable. In works on honeycomb gas seals, the practice has by now been consolidated of calling the components of H
impedances, by analogy with the electrical terminology (Dawson, Childs, et
al.; Kleynhans and Childs). However, a less confusing alternative, for a broader audience, might be to call them complex stiffness coefficients. e terms on the diagonal Hxx, Hyy are called direct coefficients, while the
terms off the diagonal Hxy, Hyxare called cross-coupled coefficients.
e continuous Fourier transform can be interpreted as a decomposition into harmonic oscillations. In combination with the decomposition along two fixed directions x, y, we can speak of a decomposition into harmonic planar oscillations. An alternative decomposition is possible, into forward and backward circular precessional orbits, also called forward and backward whirl (Bucher and Ewins). Accordingly, the forces must also be decom-posed into forward and backward rotating components. By convention, the direction of the sha rotation is the forward direction.
is decomposition can be derived from a spatial Fourier transform around the circumference. e notion is developed further in Sections.,..
e transition from one decomposition to the other amounts to a change of basis. e interpretation of ω, for a precessional motion, is that of an angu-lar speed. If we denote by the superscript (p) the decomposition into planar orbits and by the superscript (c) the decomposition into circular orbits, the change of basis can be wrien
ˆ e(p) =T ˆe(c), (.) ˆ F(p) =T ˆF(c), (.) where T = [ 1 1 ´i i ] . (.)
.. Dynamic coefficients e transformation rule for the dynamic coefficients results from that for forces and displacements
H(p) =T H(c)T´1 (.)
so that eq. (.) holds in both reference systems. e backward and forward components will be denoted by the subscripts f, b, so we can write
e(p) = [e
x, ey]T, e(c) = [ef, eb]T. (.)
Gas seals, usually, are axially symmetric and behave in the same way in every transversal direction. is behaviour is called isotropic. If at equilib-rium the rotor is eccentric, the axial symmetry is broken and the seals, in principle, are no more isotropic. e effect of eccentricity, however, is small. For honeycomb gas seals, it is usually negligible. It can be shown that, for an isotropic system, the following relations hold
Hxx = Hyy, Hxy = ´Hyx, Hf b= Hbf = 0. (.)
As a consequence, for an isotropic system we can write
H(p) = [ Hd Hc ´Hc Hd ] , H(c) = [ Hf 0 0 Hb ] . (.)
e forward stiffness and damping coefficients are usually called in liter-ature effective stiffness and damping (Kleynhans and Childs)
Keff = Kff =ℜHff, Ceff = Cff =ℑHff/ω. (.)
ey are important indicators of the performance of a seal, because, when positive, contribute to the stability of the system (and vice versa).
In Figure .and Figure.schematic representations of direct and ef-fective dynamic coefficients are shown. ey are just graphical ways to in-terpret the definitions of the dynamic coefficients, not physical models of honeycomb seals.
e most important excitation in a rotordynamic system is usually syn-chronous with the rotations of the sha, hence forward precessional. For that reason, the stability contribution of a seal is best judged by the effective coefficients.
Since forces and displacements are real variables, their Fourier trans-forms are Hermitian symmetric. e dynamic coefficients, in planar orbit decomposition, are also Hermitian symmetric, namely
Characterization of honeycomb gas seals
Figure .: Direct coefficients. Figure .: Effective coefficients.
On the contrary, in circular orbit decomposition the following relations hold
Hff(´ω) = Hbb˚(ω) (.)
Hf b(´ω) = Hbf˚(ω) (.)
In terms of stiffness and damping coefficients, eq. (.) becomes
Kff(´ω) = Kbb(ω), Cff(´ω) = Cbb(ω). (.)
e forward components, for negative values of ω, represent backward pre-cessions. Since the coupled coefficients Hf b, Hbf are null or negligible, all
the information on a honeycomb seal’s dynamic behaviour can be condensed in Hff.
ese definitions are easily extended to include the tilting movements. Instead of eq. (.) we write
´ [ˆ F ˆ M ] = [ HLL HLA HAL HAA ] [ ˆc ˆ γ ] (.) where the angles γ and the corresponding moments M have been introduced. Note that the the blocks HAL, HLAare, in general, neither equal nor
symmet-ric. Although I have not worked out an analytical counterexample, numer-ical computations with showed that they are in fact quite different. Each 2 ˆ 2 block of the dynamic coefficient matrix will exhibit the symmetry properties discussed above, according to the isotropy of the system and the chosen reference system.
. Experimental data
During my doctoral activity the (Ultra High Pressure) test rig of GE Oil & Gas, Florence, was used to perform several tests on smooth-rotor
honeycomb-.. Experimental data stator seals, mainly with diameter mm and length mm. e ranges of values for the test parameters are given in Table..
upstream pressure pU = 30to 200 bar
pressure ratio pD/pU = 0.3to 0.9
upstream temperature TU » 300 K
rotational speed fΩ= 10, 000to 14, 000 rpm preswirl ratio Rsw» ˘1
Table .: Test parameters.
In Table . a comparison is shown between the testing parameters of tests with the most comprehensive collection of experimental data I have found in literature (Dawson and Childs). In the table, Res = 2aρcs/η
is the Reynolds number of sound speed. e seals were shorter and the Reynolds numbers involved were larger, due to higher pressures.
Dawson & Childs
L/D 0.3 0.75 a/R 0.0033 0.0033 b/a 6.36 12.2 pD/pU 0.3to 0.9 0.4to 0.5 uR/cs 0.3to 0.36 0.18to 0.35 Res (0.4to 2.5)ˆ106 (0.6to 1.4)ˆ105
Table .: Comparison of testing ranges in non-dimensional form.
.. Testing apparatus
A more detailed description of the testing apparatus can be found in Vannini et al.. Two identical seals are tested at once. ey are mounted in the test cell in a symmetrical back-to-back configuration, Figure.. e relevant conditions for a compressor seal are reproduced, namely the high and low pressures, respectively upstream and downstream the seals, the rotational speed of the rotor, and the swirl of the gas before entering the seals (preswirl). e gas used in the plant is nitrogen. e design pressure is bar.
e rotor is supported by Active Magnetic Bearings (AMB). e magnetic bearings can be controlled in such a way as to impose the desired displace-ments to the axis of the rotor. e relative position between the sha and each of the bearings is measured, and so is the force actuated by each bearing.
Characterization of honeycomb gas seals
Figure .: Test cell.
e orbit imposed to the rotor axis is given by the superposition of har-monic displacements at different concurrent frequencies (multi-frequency excitation).
Forces and displacements are measured in two configurations: with a pressure difference across the seals, and without pressure (baseline). In this way it is possible to separate the dynamic coefficients of the seals from those of the whole system.
In addition to the seals, also the inertia of the rotor contributes to the dy-namic coefficients of the system. is contribution can be computed through the displacements and the known mass distribution of the rotor. However, the baseline subtraction provides a more direct measure and is preferred.
e preswirl ratio, i.e. the ratio of the circumferential velocity compo-nent of the gas to the peripheral speed of the rotor, is generated through the swirler ring. It collects the gas flow from an inlet plenum and injects it toward the seals through aerodynamic nozzles evenly distributed along the circumference. e current swirler is designed to produce a high preswirl ratio „ 1 at 10 krpm.
.. Experimental data
.. Identification methodology
e measurements of forces and displacements are sampled at a sufficiently high rate to obtain a reliable spectral decomposition in the range of the ex-citation frequencies.
For each spectral component, equation (.) holds. It consists of two scalar equations, that are not enough to determine the four impedances of the general formulation. However, assuming isotropy, there are only two unknowns Hd, Hc ´ [ˆ Fx ˆ Fy ] = [ Hd Hc ´Hc Hd ] [ ˆ cx ˆ cy ] (.) e unknown impedances can be put in evidence by rearranging the terms
[ ˆ x yˆ ˆ y ´ˆx ] [ Hd Hc ] = ´ [ˆ Fx ˆ Fy ] (.) e shape of the orbit of each selected frequency should not be circular or near to circular, because the matrix of coefficients in equation (.) would become ill-conditioned. Planar orbits are preferred with this identification method.
To obtain the four impedances of the general case, two experiments are necessary, differing only for the excitation components and not for pressures, temperature and preswirl. Strict acceptability criteria must be met to couple two experiments. For the two coupled experiments, we can write
´ [ˆ F1x Fˆ2x ˆ F1y Fˆ2y ] = [ Hxx Hxy Hyx Hyy ] [ ˆ c1x ˆc2x ˆ c1y ˆc2y ] (.) which is immediate to solve.
Chapter
One-dimensional honeycomb
seal models
. Introduction
e first effective way to model honeycomb annular gas seals has been the bulk flow model devised by Kleynhans and Childs. It was the evolution of a bulk flow model used by Nelsonbfor plain annular seals. e model presented by Nelson was, in turn, inspired by those used by Hirs for turbulent lubricant films and by Zuk et al.for face gas seals.
e dynamic behaviour depends, of course, on the dynamics of the fluid traversing the seal. e fluid passes through a narrow clearance, whose ge-ometry varies with time due to the vibrations of the solid parts. e clearance height is so small, with respect to the overall dimensions of the seal, that the geometry of the clearance can be effectively described by means of a refer-ence surface and a thickness.
e basic idea of the bulk flow models is to assume that, for each point of the reference surface, it is sufficient to know the dynamics of the flow in the middle of the thickness, i.e. the clearance, to get an approximate description of all the flow. In fact, bulk flow models assume that the flow velocity, density and temperature vary lile across the clearance, apart from thin layers near the containing walls.
e dimensionality of the problem can further be reduced from two to one, when the axial symmetry of the seals is considered. e circumferential variations can be discretized by means of Fourier series. Since the excitation is harmonic around the circumference, the Fourier series is particularly sim-ple and reduces to two terms, describing forward and backward travelling
One-dimensional honeycomb seal models waves.
e bulk flow assumption can be weakened, as there is no need to sup-pose a constant profile of velocity and fluid variables across the channel. It is only needed that the profile can be described with a finite number of pa-rameters. e expression “one-dimensional model” will be used here in the broader sense, and “bulk-flow model” in the stricter sense. Already Zuk et al.
used the expression “quasi-one-dimensional model”.
e bulk-flow models for gas seals have the same disadvantages as the bulk-flow models used for pipes. First of all, they rely on empirical relations to model the interactions between fluid and walls. Another delicate point is the treatment of the inlet and exit regions. Bulk-flow models can do a good job when the flow can be assumed fully developed, i.e. “sufficiently far” from the ends.
While for pipes there is a consolidated corpus of validating data, for our seals the situation is much less fortunate. One can try to adapt the empirical relations for friction and heat transfer but, besides the geometric difference, there are other important factors that require novel extensions. e flow is compressible, and constitutes a challenge even for pipe flow. e rotation of the sha induces, in combination with the axial pressure-driven flow, a cir-cumferential Couee flow. And finally, the wall interactions can be sensitive to vibrations.
In this work we are dealing with annular seals, whose reference surface is cylindrical. In the following section, a one-control-volume model for plain annular gas seals, similar to that of Nelson b, is described. e two-control-volume model of Kleynhans and Childsfollows. In discussing these models, some possible variants are pointed out. ese variants have been implemented in the code , described subsequently.
. One-control-volume bulk flow model
is model is called one-control-volume to distinguish it from the two-control-volume model, described later, where a new control two-control-volume has been added to model the honeycomb cavities.
e problem is solved in steps. e first step consists in finding the steady flow, with the sha of the rotor fixed in a centred position. en the dy-namic behaviour is studied by imposing precessional circular orbits to the sha. is step is handled as a small perturbation problem. Finally, the re-acting forces are computed by integrating the pressure distribution, and the dynamic coefficients are computed according to their definition. e rotor
.. One-control-volume bulk flow model and the stator are assumed to be rigid bodies, and the rotor is assumed to be fixed. Since the inertia of the fluid is very small, only the relative displace-ments of rotor and stator are relevant.
e governing equations are balance equations of mass, momentum and energy. At this stage, the problem is two-dimensional, with independent variables θ, z respectively circumferential and axial coordinates. e depen-dent variables describe the flow in the bulk. We use here, as dependepen-dent vari-ables, density ρ, velocity u, and temperature T . e velocity is here a two-dimensional vector, because the radial component is null. e equations can of course be wrien with a different choice of dependent variables, the most common alternative being to use the pressure p instead of the density.
e balance equations have all a common form. For a generic extensive quantity ϕ, the balance can be stated as
Bϕ
Bt +div(Dϕ+ ϕu) = σϕ (.)
where σϕis the source or production term, Dϕis the diffusion term and ϕu
is the convection term. Convection and diffusion are the two possible trans-port mechanisms, so their sum is the transtrans-port term. e balance equation calculates the local increase of ϕ as a balance of production and transport.
e walls are impermeable, so that in our problem there are no mass sources. Momentum and energy, on the contrary, can cross the wall bound-aries and provide source terms to our balance equations. Momentum dif-fuses through contact forces, namely pressure and viscous stresses. Energy diffuses in two ways, through heat conduction and work.
e balance equations for our problem can be wrien as (aρ),t+ (aρuz),z+ (aρuθ),θ/r = 0
(aρuz),t+ (aρu2z),z+ (aρuθuz),θ/r + ap,z = τz
(aρuθ),t+ (aρuθuz),z + (aρu2θ),θ/r + ap,θ = τθ
(aρet),t+ a,tp + (aρhtuz),z+ (aρhtuθ),θ/r = q + uRτRθ
(.)
where a is the clearance height, r the seal radius, uRthe peripheral speed of
the rotor, etthe total energy, htthe total enthalpy. e source terms on the
right describe the interaction with the walls and must be wrien as functions of the dependent variables by means of empirical relations. ey are: the forces per unit area that the walls exert on the fluid, τR, τS, whose sum is τ ;
and the heat per unit area that they release to it, qR, qS, whose sum is q. e
One-dimensional honeycomb seal models the dependent variables through the equation of state and other constitutive equations of the gas.
It can be noted that most diffusion terms are null in eq. (.). ere is no mass diffusion, and viscous stresses and conduction in θ, z directions are considered negligible, so that momentum and energy diffuse only through terms related to the pressure.
To complete the two-dimensional model, boundary conditions must be set at the edges of the domain, i.e. at the entrance and at the exit. e boundary conditions on the two boundaries are qualitatively different. It can be demonstrated that perturbations of swirl uθand temperature T
can-not travel upstream. is is due to the absence of viscous and conductive terms. As a consequence, swirl and temperature can only be imposed at the entrance. e two remaining relations are obtained by imposing the pressure on both boundaries.
ere is one exceptional case, because the seals can operate in choked condition, with fluid flowing at sound speed at the exit. In this case, no per-turbation at all can propagate upstream, not even pressure, and the boundary condition on the exit pressure must be restated as a condition on the axial ve-locity. e wall models and the boundary-condition models will be discussed in later sections.
e steady problem reduces immediately to a one-dimensional model, by eliminating the dependence from θ and t
(aρuz)1 = 0
(aρu2z)1+ ap1 = τz
(aρuθuz)1 = τθ
(aρhtuz)1= q + uRτRθ
(.)
is is a non-linear ODE system that can be rearranged in matrix form as
A(y, z) y1 =a(y, z) (.)
where the dependent variables form the vector
y = [ρ, uθ, uz, T ]T. (.)
Together with the boundary conditions at inlet and exit
cI(yI) = 0, cE(yE) = 0, (.)
where yI, yE are the dependent variables evaluated at the respective
bound-aries, this constitutes a boundary value problem that can be solved numeri-cally.
.. One-control-volume bulk flow model e perturbation equations can be derived as follows. First of all, a first-order perturbation of the two-dimensional differential equations eq. (.) and of the accompanying boundary equations must be wrien. is step can be formalized by introducing a perturbation operator in the following way. Let us use a circle over the leers to indicate perturbed variables, no additional diacritic to indicate unperturbed variables, and a breve sign over the leers to indicate the perturbation, so that the perturbed dependent variables are
8 ρ = ρ + ˘ρ 8 uθ= uθ+ ˘uθ 8 uz = uz+ ˘uz 8 T = T + ˘T (.)
and the perturbed clearance height is 8
a = a + ˘a (.)
e breve sign serves as the perturbation operator and can be applied to any expression to indicate the difference between the perturbed and unper-turbed evaluation of the same expression, for example
(aρuz)˘ = (8a 8ρ 8uz) ´ (aρuz) (.)
e perturbation operator is actually a differential operator and the usual differentiation rules apply
(aρuz)˘ = ˘aρuz+ a ˘ρuz+ aρ˘uz (.)
e two-dimensional perturbation equations can thus be simply wrien as the perturbation of eq. (.)
(aρ)˘,t+ (aρuz)˘,z + (aρuθ)˘,θ/r = 0
(aρuz)˘,t+ (aρu2z)˘,z+ (aρuθuz)˘,θ/r + (ap,z)˘ = ˘τz
(aρuθ)˘,t+ (aρuθuz)˘,z+ (aρu2θ)˘,θ/r + (ap,θ)˘ = ˘τθ
(aρet)˘,t+ (a,tp)˘ + (aρhtuz)˘,z + (aρhtuθ)˘,θ/r = ˘q + (uRτRθ)˘
(.)
e time coordinate t is transformed into the angular frequency ω by a continuous Fourier transform. e equations become one-dimensional by transforming also the θ coordinate into a discrete coordinate k, the wave number, by means of a Fourier transform. Since the excitation ˘a is harmonic
One-dimensional honeycomb seal models in θ with full period 2π, only two terms of the Fourier series are non-null, forward (k = ´1) and backward (k = +1). e forward and backward components are not independent, it is sufficient to solve one of them, and we focus on the forward solution. e one-dimensional perturbation equations are thus
iω(aρ)˘ ´ i(aρuθ)˘/r + (aρuz)1˘ = 0
iω(aρuθ)˘ ´ i(aρu2θ)˘/r + (aρuθuz)1˘ ´ ia˘p = ˘τθ
iω(aρuz)˘ ´ i(aρuθuz)˘/r + (aρu2z)1˘ + (ap1)˘ = ˘τz
iω(aρet)˘ + iω˘ap ´ i(aρhtuθ)˘/r + (aρhtuz)1˘ = ˘q + (uRτRθ)˘
(.)
e variables in eq. (.) are of course Fourier transforms, even though this has not been explicitly indicated in notation. Eq. (.) constitute a linear ODE system that can be rearranged in matrix form as
A ˘y1+B ˘y = b = ˘a b
0+ ˘a1b1 (.)
Note that matrix A is the same as in the unperturbed eq. (.). e linear boundary value problem is completed by the perturbed boundary conditions
CI˘yI= 0, CE˘yE = 0. (.)
In the perturbation problem, the value of the perturbed clearance height ˘
ais imposed. If cf is the forward component of the displacement, as defined
in Section., then the relationship is quite simple ˘
a = ´ˆcf (.)
To include the tilting movements, we can write ˘
a = ´ˆcf ´ ˆγfz (.)
e angular displacements γxis a rotation around the axis y such that the axis
xmoves toward z, and analogously for γy. e rotating force and rotating
moment are obtained by integration ˆ Ff = ´π żzE zI (˘p + i˘τRθ) dz, (.) ˆ Mf = ´π żzE zI (˘p + i˘τRθ)z dz. (.)
e effect of the friction stress τRθis negligible for gas seals.
As a final note, an isothermal model is obtained by leaving out the energy equation.
.. Two-control-volume bulk flow models
. Two-control-volume bulk flow models
In Figure.the two control volumes are shown. Volume A is the clearance and volume B comprises the cavities of the honeycomb, or the holes of the hole paern.
Figure .: Control volumes.
It is assumed that the fluid in the cavities has negligible momentum. However, fluid can freely be exchanged between the two control volumes. e balance equations can be established for the two volumes together. In comparison with eq. (.), only the time-derivative terms of the mass and energy balances change
(aρ),t+ bρB,t+ (aρuz),z + (aρuθ),θ/r = 0
(aρuz),t+ (aρu2z),z + (aρuθuz),θ/r + ap,z = τz (.)
(aρuθ),t+ (aρuθuz),z+ (aρu2θ),θ/r + ap,θ = τθ
(aρet),t+ b(ρBeBt),t+ a,tp + (aρhtuz),z + (aρhtuθ),θ/r = q + uRτRθ
e mean hight b of the cavities is defined as the ratio of the volume of a cavity to the area of a hexagon of the paern.
e fluid properties in volume B can be described using three variables. To write them as functions of the fluid properties in volume A, three addi-tional equations must be established. I see no reason to include spatial gra-dients of the flow variables in these equations. is means that the number of differential variables and equations does not change.
One equation is simply pB = p, based on the radial equilibrium. e
two remaining relation are less obvious, and must be establish on empirical grounds. For an isothermal model, the kinetic energy is irrelevant and the density is ρB = ρ.
One-dimensional honeycomb seal models
.. Boundary conditions
In the upstream “reservoir” the fluid has negligible axial velocity, but the circumferential velocity can be significant. e upstream conditions involve
pU, TU, and uU θ. From the downstream “sump”, only the pressure pDhas an
effect on the seal flow, because heat and swirl momentum cannot propagate upstream. e entrance and exit boundaries of the seal will be denoted by the subscripts I, E.
e entrance conditions are derived from the balance of momentum and energy, while only the axial momentum is involved in the exit conditions. Two of the entrance conditions are simply
uU θ= uθ|I (.)
hU t = ht|I (.)
e conditions on pressure are expressed in terms of loss coefficients, kI,
kE, assigned by the user. e loss coefficients are the ratio of the amount of
energy transformed irreversibly into heat to the kinetic energy in the duct. For a perfect gas, assuming an adiabatic transformation, the following bound-ary conditions result (Nelsonb)
pU = p (1 + mξIu2z/c2)1/m|I ξI= (1 + kI)/2 (.)
pD = p (1 + mξEu2z/c2)1/m|E ξE = (1 ´ kE)/2 (.)
m = (γ ´ 1)/γ c2 = RgT
e entrance loss coefficient should be near to 0, i.e. no loss, while the exit loss coefficient should be near to 1, i.e. no pressure recovery.
In the isothermal case, it is legitimate to take the limit for m Ñ 0
pU = pexp (ξIu2z/c2)|I (.)
pD = pexp (ξEu2z/c2)|E (.)
For an incompressible fluid, both conditions become
pU = p + ξIρu2z|I (.)
pD = p + ξEρu2z|E (.)
is formulation can be viewed as an approximation for low velocities, be-cause compressibility effects become less important. In fact, the axial Mach
.. Friction model number at the entrance is usually less than 0.3 and the incompressible con-ditions are a justified approximation. In both compressible and in-compressible conditions are used, in only inin-compressible.
Care must be taken to recognize a choked flow. e axial velocity uz
increases with the axial coordinate, as in a Fanno flow (Shapiro). Unless the clearance is sufficiently divergent, which usually does not happen, the flow cannot reach the critical velocity at any point along the seal, except at the exit. When the choked condition is reached, the flow is insensitive to the downstream pressure pDand the exit condition must be replaced by
Maz|E = 1 (.)
where Mazis the axial Mach number, defined as
Maz= uz/cs (.)
for the non isothermal case, and as
Maz = uz/c (.)
for the isothermal case. e sound speed differs for the two cases
cs=aγRgT , c =aRgT . (.)
e discontinuity of the two formulations calls to mind a paradox. It should be possible to pass, with a continuous transformation, from an adi-abatic to an isothermal solution, by changing continuously a heat transfer coefficient from zero to infinity. If there is a point at critical conditions, the non-isothermal solution cannot converge pointwise to the isothermal one. It can, however, converge almost everywhere.
. Friction model
Only a friction model based on Blasius’ formula has been implemented in . A comparison of different friction models was made by D’Souza and Childs. Let v be the velocity of the fluid relative to the wall, namely
vR=u ´ [0, uR]T, vS =u. (.)
To model anisotropic friction, a matrix of friction coefficients f can be defined such that
One-dimensional honeycomb seal models As principal directions of the friction matrix have been chosen, for simplicity, the axial and circumferential directions, and the Blasius’ formula has been implemented as f = Rem [ nz 0 0 nθ ] . (.)
e Reynolds number in this formula is defined as Re = 2aρv
η . (.)
Friction on the rotor is assumed isotropic, with mRz= mRθ= mR.
Sen-sible values for mR, nR, based on the pipe flow, can be found in the manuals.
For the stator, there is no simple predictive model. Experimental values for an isotropic model were obtained on a flat plate tester (Ha and Childs) for some geometries and Reynolds numbers.
Blasius’ formula has been validated for steady and fully developed flows. e extension to unsteady flows presents subtle difficulties. Perturbations at a given non-null frequency might behave differently, because the periodic oscillations of the viscous layer might have a significant influence on the transfer of momentum to the wall.
In other words, we can model the dynamical behaviour of friction with an empirical correlation that depends on frequency, but we only have exper-imental support, so far, for zero frequency.
e bulk flow equations obtained in Section.differ from those in Kleyn-hans’ dissertation (Kleynhans), although in all appearance based on the same principles. is is a good example of the pitfalls involved in extending steady empirical correlations to unsteady problems.
In Kleynhans’ dissertation, the balance of momentum is operated on con-trol volume A alone, i.e. on the channel. e momentum exchanged with the stator is thus the composition of two contributions: one of them is based on the Blasius’ formula, the other on the mass exchange between control vol-umes. Let the mass flow from vol. A to vol. B be
ϕρ= b ˘ρ,t. (.)
When the flow is positive, the fluid that enters the cavity yields immediately its momentum to the stator. Conversely, when the flow is negative, the fluid that exits the cavity is sucked in the channel, drawing the needed momentum from the stator. is additional momentum exchange is estimated to be
.. Heat transfer model is model is sensible, but is anyway a wild guess. To check the impor-tance of this momentum transfer contribution on the dynamic coefficients, a linear interpolation has been done, in , of the two models. A factor cf
has been introduced, controlled by the user. e resulting “dynamic” friction stress is
˘
τ = ˘τf ´ cfb ˘ρ,tu (.)
where ˘τf is the part of momentum transfer based on the steady Blasius’
for-mula.
. Heat transfer model
e most important phenomenon that the heat equation and the heat transfer model should be able to describe is the amount of heat produced by friction and how much of it is transferred through the walls. e effect of the rate of heat transfer on the speed of sound is also important. e temperatures of rotor and stator are assumed to be known and equal to the temperature of the fluid upstream.
For the heat transfer, a simple correlation has been posited between Nus-selt and Reynolds numbers,
Nu = nqRemq (.)
where mq = 0.8. is correlation can be viewed as a simplification of
Sieder-Tate correlation
Nu = 0.027 Re4/5Pr1/3(η/η
w)0.14 (.)
where the dependence of viscosity, thermal conductivity and heat capacity on pressure and temperature have been neglected.
An adiabatic wall temperature Taw is introduced, as defined by Shapiro , to take into account the fact that, for a compressible fluid, the temper-ature in the bulk differs from the tempertemper-ature in the viscous layer near the wall, even when the wall is adiabatic. e adiabatic wall temperature differs a lile from the stagnation temperature, and their ratio Rw depends mainly
on Prandtl number. Since Prandtl number varies lile for air, nitrogen, and other fluids of interest in compressors, a fixed value Rw= 0.9was used. e
adiabatic wall temperature is thus calculated as
Taw= Rw ( T + v 2 2cp ) . (.)
One-dimensional honeycomb seal models e heat transfer with the wall is computed as
q = λ (T ´ Taw), with λ =
Nu κ
2a . (.)
In the limit case with nq= 0, the flow is adiabatic. In the limit case with
nqÑ 8 the flow is not isothermal, because what remains constant and equal
to wall temperature is the adiabatic wall temperature. e bulk temperature is not constant, but decreases with velocity.
Connected to this heat transfer model, is the question about how to de-termine TB. In the kinetic energy is neglected and the temperature
in the cavity is defined as
TB= wT (.)
where the factor w is controlled by the user through a parameter cwand has
the value w = cwRw ( 1 + u 2 2h ) . (.)
. Numerical methods
.. Unperturbed equationsIn Section.the unperturbed problem has been formalized as a boundary value problem
A(y, z) y1 =a(y, z) (.)
cI(yI) = 0, cE(yE) = 0 (.)
A boundary value problem can either be solved indirectly, with a shoot-ing method, or directly. A shootshoot-ing method consists in transformshoot-ing the problem in an initial value problem, using only the conditions on one side. e necessary additional conditions are introduced as unknown parameters, and the initial condition problem is solved iteratively until the conditions on the other side are met.
In our problem, we can impose the mass flow rate as a parametric initial condition. A Runge-Kua integration scheme can be used for the initial-value problem and the non-linear equation for the exit condition can be solved, for example, with a bisection method. Imposing the mass flow rate
ϕ = aρuzhas an additional advantage, in that it is already an integral of the
problem and the mass balance equation can be dropped, together with one dependent variable, e.g. uz.
.. Numerical methods e trouble with this approach is that, near choking conditions, the prob-lem is ill-conditioned. Small variations in mass flow rate induce large varia-tions of the downstream pressure.
However, the mathematical problem, in itself, is expected to be well-conditioned, because the boundary conditions are well posed on physical grounds. Small variations of each of the four boundary conditions should produce effects of comparable magnitude, even in choked or almost choked conditions. In the boundary value problem is solved by means of a collocation method, a Lobao IIIa method implemented in func-tionbvp4c. e mass balanced equation cannot be dropped with this method,
because the mass flow rate is not known in advance.
Matrix A in eq. (.) can be wrien, with some rearrangement of the balance equations, in the form
A = uz ρ 0 0 c2 ρuz 0 α/β 0 0 ρuz 0 ´c2αT uz 0 0 ρcvuz (.)
e determinant vanishes when uz = 0and when uz = cs(for the isothermal
case uz= c). us, as the axial Mach number approaches 1, the spatial
gradi-ents become arbitrarily large. is behaviour recalls a convergent/divergent nozzle, and in fact the equations are the same, with the addition of swirl, that does not change the qualitative behaviour.
If the clearance height is constant or convergent, the critical conditions can be reached only at exit. In the exiting jet, shock waves form, through which the velocity and pressure of the fluid are adjusted to the sump condi-tions. When the clearance height is divergent, the expansion due to friction (Fanno flow) combines with the contraction due to the geometry, so that it is possible for the critical section to be at an intermediate place in the seal. at would be a complication, because shock waves could form in the inte-rior of the seal. It is not a realistic scenario, as in real cases the friction effect prevails. From a numerical point of view, however, it is important that the determinant never changes sign during the solution process.
Note that the determinant vanishes also for uz = 0. is is connected
with the fact that, with a reversed flow, the boundary conditions too must be reversed. In , the dependent variables have been changed to
One-dimensional honeycomb seal models where w = d Maz 1 ´ Maz, (.)
so that there is no need to impose bounds on them. Of course, the user can input geometries and values for which no solution exists, and the solver will not converge.
.. Exit boundary condition with penalty function
e exit boundary condition has been rendered continuous, with the aid of a penalty function approach. With penalty function
ψ = 1 ´ Manzp (.)
the boundary condition is wrien as
pD = pmψ|E (.)
where pmdepends on the model chosen, respectively non-isothermal (.),
isothermal (.), and incompressible (.)
pm= p (1 + mξEu2z/c2)1/m (.)
pm= pexp (ξEu2z/c2) (.)
pm= p + ξEρu2z (.)
.. Artificial volume viscosity
In the effort of finding well-conditioned formulations of the flow equations, a side problem has been solved, that is worth mentioning. For some shapes and conditions, supersonic flow and normal shock waves can develop in the interior of the seal. Even if the existing honeycomb gas seals are far from these conditions, it might be useful to deal with possible shock waves while numerically experimenting new solutions.
It must be said that lile is known of the interaction of a supersonic wave with the honeycomb cavities. is model can only be considered a starting point.
If the shock region is thin, it is not necessary to know exactly what hap-pens in its interior to solve the problem. A zero-thickness shock-wave prob-lem is well defined, and the solution is independent of the dissipative mech-anism involved.
.. Numerical methods e dissipation mechanism has been chosen to be a volume viscosity. e thermal conductivity has been discarded because it is not applicable to an isothermal model, and the dynamic viscosity because it leads to a more complex formulation.
One variable, some new terms, and one equation are added to eq. (.). e new equations are
(aρ),t+ bρB,t+ (aρuz),z+ (aρuθ),θ/r = 0
(aρuθ),t+ (aρuθuz),z + (aρu2θ),θ/r + ap,θ´ aζv,θ/r = τθ
(aρuz),t+ (aρu2z),z+ (aρuθuz),θ/r + ap,z´ aζv,z = τz
(aρet),t+ b(ρBeBt),t+ a,tp + (aρhtuz),z+ (aρhtuθ),θ/r +
´ a,tζv ´ ζ(vuz),z ´ ζ(vuθ),θ/r = q + uRτRθ
v = (auz),z+ (auθ),θ/r + a,t/a
(.)
where v is the divergence of velocity and ζ the volume viscosity. With some rearrangement of the equations, we have now
A = uz ρ 0 0 0 c2 ρuz 0 α/β ´ζ 0 0 ρuz 0 0 ´c2αT uz 0 0 ρcvuz 0 0 ρ 0 0 0 (.)
whose determinant vanishes only for uz= 0.
e value of the volume viscosity ζ must be chosen small enough to con-fine the shock to a narrow region, but not too small, to avoid numerical dif-ficulties. In practice, a wide range of values is appropriate. A solution can be refined iteratively, if needed. e volume viscosity can be varied from one analysis to the other, while the previous solution can be used as a starting point for the next computation.
One additional boundary condition is needed. It is possible to impose
v = 0at inlet. If ζ is chosen sufficiently small, the solution is sensitive to
the imposed value only for a short length downstream. e exit boundary condition must also be changed. Instead of the penalty function formulation of eq. (.), the following relation is used
pD = pm´ ζv|E. (.)
e model with volume viscosity has been implemented in for the isothermal model alone.
One-dimensional honeycomb seal models
.. Perturbed equations
e perturbed equations are linear and present no additional numerical dif-ficulties. e same solver have been used as for the unperturbed equation. e main implementation difficulty has been the analytical differentiation implied in the small perturbation theory.
. code
e one-dimensional models described in the previous sections have been implemented in a computation tool, named (Hole Paern SEAL). e computation algorithms have been coded in and are not optimized for computational speed. e main concerns were development speed and modularity.
In Figure.the structure of the program is shown. e modules repre-sented with thickened boundaries implement parts of a bulk-flow model that are, to a large extent, independent. In fact, the internal interfaces have been designed in such a way that different implementations of each module can be plugged in interchangeably.
Figure .: structure.
e physical variables share common conventions through all the mod-ules, except the solver, where some transformations of the variables are
use-.. Results ful for numerical reasons. ese variable transformations braught some com-plication to the interface, as can be seen by the presence, in Figure., of “translator” modules.
e following modules were implemented and used to compare numeri-cal and experimental results:
- Bulk module: isothermal, non-isothermal;
- Boundary-condition module: compressible, uncompressible;
- Wall-interaction module: implements Sec.. for friction and Sec. .
for heat exchange.
Some variants of the models are implemented through the use of param-eters that can vary in the range 0 to 1, namely
- cf (eq..): dynamic contribution of cavities to friction;
- cw(eq..): rise of temperature in cavities.
e model implemented by the numerical code is reproduced by the isothermal model of with incompressible boundary conditions and cf = 1.
e input to the main program consists in: - operating point;
- geometry;
- choice of submodels with their options and parameters; - initial guess;
- requested output.
e operating point is defined by the operating parameters pU, TU, uU θ,
pD, Ω. e geometry is input by assigning a, b, r, z as functions of a
paramet-ric coordinate s. ey can be input as splines. us, the seals can have any revolving shape, not necessarily cylindrical. However, the bulk-flow engines implemented so far assume constant b and r. Each submodel needs its own parameters, for example the friction parameters. e initial guess can be the result of a previous simulation, otherwise it is generated by the program. e user can request a steady analysis or an unsteady analysis at frequency ω.
e output contains the solutions to the systems, the mass flow rate and, for the unsteady simulations, the dynamic coefficients.
. Results
e decision of developing a new prediction tool, similar to but more flexible, was taken before the first results were available from the
One-dimensional honeycomb seal models test rig. We wanted to be prepared to refine and tune the model and achieve the maximum possible agreement with the experimental data.
When, in the summer of , the first results for honeycomb gas seals were obtained, the difference between experiments and predictions was found to be greater than expected. e new variants introduced in did not solve the problem. On the contrary, they predicted dynamic coefficients surprisingly similar to the old ones, and le the experimental data unexplained.
In Figure.and Figure.experimental data from three different tests at different pressures are shown, in non-dimensional form. e non-dimensional comparison is interesting, because it shows similarity of behaviour. Up-stream and downUp-stream pressures [bar] are shown in the legend. e pres-sure ratios are close to each other, pD/pU = 0.79 to 0.83. e forward
preswirl uU θ/uR is near to 1. Angular frequency and stiffness are
non-dimensionalized respectively with
ωref = c/R (.) Kref = πRLpU a + b (.) −0.4 −0.3 −0.2 −0.1 0 0.1 0.2 0.3 0.4 0.13 0.14 0.15 0.16 0.17 0.18 0.19 0.2 0.21 0.22 0.23 frequency Re(Hff) 31/26 64/51 127/102 191/150
Figure .: Forward and backward impedances in non-dimensional form for similar pressure ratios, real part. Entrance and exit pressures [bar] are shown in the legend. A distinctive feature, recognizable only in experiments with comparable pressure ratios, is the rapid change of effective stiffness Keff = ℜHff, from
backward to forward precessions. is feature is unaccounted for, but shows good repeatability.
.. Results −0.4 −0.3 −0.2 −0.1 0 0.1 0.2 0.3 0.4 −0.8 −0.7 −0.6 −0.5 −0.4 −0.3 −0.2 −0.1 0 0.1 0.2 frequency Im(Hff) 31/26 64/51 127/102 191/150
Figure .: Forward and backward impedances in non-dimensional form for similar pressure ratios, imaginary part. Entrance and exit pressures [bar] are shown in the legend.
In Figure.and Figure.the experimental results for the 64/51 case are compared with four simulations. For all the simulation the loss coefficients are kI = 0, kE = 1and the boundary conditions are of incompressible type.
e friction parameters of the rotor are also the same. Other parameters are shown in Table..
If the value of the coefficient of thermal exchange nqis not indicated, then
the simulation is isothermal. For non-isothermal simulations, it is cw = 0,
i.e. the temperature in the cavity is the same as in the channel.
−0.5 −0.4 −0.3 −0.2 −0.1 0 0.1 0.2 0.3 0.4 0.5 0.1 0.15 0.2 0.25 0.3 0.35 frequency Re(Hff) isot momf isowall adiab exp
Figure .: Forward and backward impedances in non-dimensional form, real part, unchoked case.
One-dimensional honeycomb seal models −0.5 −0.4 −0.3 −0.2 −0.1 0 0.1 0.2 0.3 0.4 0.5 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 frequency Im(Hff) isot momf isowall adiab exp
Figure .: Forward and backward impedances in non-dimensional form, imaginary part, unchoked case.
label fp nq
isot 1
-momf 0
-adiab 1 0
isowall 1 8
Table .: Simulation options for comparison with experimental data. e friction factor parameters for the stator mS, nS are one of the
diffi-culties of honeycomb seal modelling. e reported simulations have mS= 0
and nSfied to reproduce the experimental mass flow rate. In this way,
er-rors on mass flow rate prediction are not cumulated with erer-rors on dynamic prediction. e choice of mShas lile influence on the dynamic coefficients.
In Figure.and Figure.the experimental results for a 49/15 case are compared with their simulations. e seal is working in choked conditions.
e differences among the models are more evident for the choked case. is is probably due to greater differences in the steady state solution. In both cases, the adiabatic model predicts a temperature rise up to K in the bulk, due to friction.
ere are many unexplained differences between models and experiments, but it is possible to establish a scale of priorities. Seals in choked conditions are expected to be more difficult to predict, because the empirical wall inter-actions might depend on Mach number, and it is possible that oblique shock waves are forming near the exit. It is desirable to obtain good predictions at least for low Mach numbers.
.. Results −0.5 −0.4 −0.3 −0.2 −0.1 0 0.1 0.2 0.3 0.4 0.5 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 frequency Re(Hff) isot momf isowall adiab exp
Figure .: Forward and backward impedances in non-dimensional form, real part, choked case. −0.5 −0.4 −0.3 −0.2 −0.1 0 0.1 0.2 0.3 0.4 0.5 −1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 frequency Im(Hff) isot momf isowall adiab exp
Figure .: Forward and backward impedances in non-dimensional form, imaginary part, choked case.
e imaginary part of stiffnessℑHff is more important for stability
con-siderations than the real part. Moreover, it can effectively by approximated by a straight line. e frequency, for which ℑHff vanishes, is known to
depend almost exclusively on swirl, hence on the unperturbed model. More important of all, is the prediction of the slope ofℑHff, especially
for unchoked conditions. Both simulation and experiment show that the slope is about the same for all the frequencies of interest.
e experimental slope is, for the unchoked case shown here, about . times steeper than that of the models so far proposed. For other unchoked cases the ratio can vary, but the experimental slope is always steeper. is difference is difficult to account for. In my opinion, this is the first check that
One-dimensional honeycomb seal models a new tentative model should pass.
Chapter
CFD model of a single cell
. Introduction
One-dimensional models, such as the bulk-flow models described in the pre-ceding chapters, rely on strong simplifying assumptions, that are difficult to validate. e details of the flow are not known from experiment and the val-idation by means of the experimental dynamic coefficients is a very indirect one.
CFD three-dimensional simulations rely on much less assumptions than one-dimensional models, although, strictly speaking, they still rely on some empiricism to model turbulence. ey are used as an effective predictive tool in an astounding range of applications.
e use of CFD simulations to predict the dynamic coefficients is possible in principle, but difficult in practice, because of the complexity of the geom-etry. e modelling of a whole seal would be a tremendous undertaking. For example, one of the seals recently tested on the test rig of GE Oil& Gas in Florence contains ˆ . = , cells ( circumferentially, and or , alternating, axially).
In principle, it is possible to exploit the circumferential periodicity. Fig-ure.shows two typical periodic sectors, depending on the orientation of the cells. e rotor excitation must be a circular precession, as for the one-dimensional models. e periodic condition consists in imposing the same flow on the two sides of the sector, but with a time shi. Continuing with our example, if we impose on one side the same flow that the other side had at the preceding time step, each excitation cycle will be divided in as many time steps as the number of periodic sectors, i.e. for the GE seal. If we impose a difference of two time steps, the excitation cycles will be divided in
. CFD model of a single cell twice as many time steps, i.e. , and so on.
Figure .: Periodic sectors of a honeycomb seal.
To my knowledge, a CFD analysis of this kind has not yet been ventured. is is probably due to the fact that commercial general-purpose CFD pack-ages do not offer this kind of periodic boundary conditions out of the box. e feasibility of this approach depends on how a candidate commercial code exposes the relevant computational structures to make non-standard exten-sions.
No other symmetries are exploitable to solve directly the fluid dynamics problem as a whole. Yet, it is thinkable to use simplified models for the en-trance and exit regions and solve separately the flow in the central part of the seal, where it can be considered as a dynamically fully developed flow.
is approach would be straightforward if we were dealing with an in-compressible fluid, because an inin-compressible fluid flow does not depend on the absolute pressure, but only on pressure gradients. us, the subproblem would be axially periodic. In our case, however, compressible effects are im-portant, and there is no way to exploit the axial geometric periodicity of the honeycomb paern.
Instead of using CFD to solve directly the whole fluid dynamics prob-lem, it is possible to set up a two-stage computational process, where a one-dimensional model is supported by CFD analyses of a single cell. e CFD analyses of a cell can at least mitigate the arbitrariness of the empiricisms of the one-dimensional model, by providing a more direct way of validating them.
e feasibility of this approach has been tested by devising and imple-menting a suitable single-cell CFD model with the commercial package . In the following pages the method and some preliminary results will be exposed.
.. Dual-scale analysis
. Dual-scale analysis
e foundation of the theoretical frame is the observation that the flow is expected to vary lile from one cell to the neighbouring one, for the very reason that the cell size is small by comparison to the seal size, and also by comparison to the length of pressure waves at the frequencies of interest.
In this sense, we can speak of a quasi-periodic solution and decompose it into fast and slowly varying components, see Figure.. At the cell scale, the slowly varying component can be substituted with a first order approx-imation, and the fast varying component with a periodic one. In the CFD simulation of the single cell, the slowly varying solution will be imposed, and the result of the analysis will give the periodic component.
Figure .: asi-periodic solution and its slow-varying component. At the seal scale, the one-dimensional ODE model is based on balance equations. e empirical relations needed to tune up the one-dimensional model can be validated and refined by examining the CFD solutions. For example, it is possible to assess the rate of momentum flux between fluid and walls as a function of the slowly varying component of the solution. In other words, it is possible to validate and improve the friction function, which, at the seal scale, is a function of the dependent variables of the ODE system and their first derivatives.
. asi-periodic assumption
Figure.shows the geometry of a cell. e dark part represents the fluid, while the light transparent parts are the rotor (below) and the stator (above). Most of the fluid is contained in the stator cavity. e six lateral faces of the prism form the periodic boundaries of the cell, and comprise both fluid and
. CFD model of a single cell solid interfaces.
Figure .: Cell geometry. Figure .: Periodic hexagon. Each point of the periodic boundary has at least a corresponding point on the opposite face, as illustrated by points A, B in Figure., that correspond respectively to points A1, B1. e same figure shows also the limiting case of a point C on the edge, that has two corresponding points C1and C2.
To formalize the quasi-periodic model, we decompose the quasi-periodic solution into a periodic part and a dri. e dri is a first order approxima-tion of a generic slowly varying funcapproxima-tion of coordinates. ere is no need for the slowly varying part to vary at all across the fluid in radial direction, so it will be a function of only two coordinates. erefore, since the dri is a first order approximation, it will be a fixed suitable function of two coordinates, depending on scalar parameters that serve to specify the gradients.
In the reference system used for the cell, x is the axial direction, y the ra-dial direction, and z the circumferential direction. See also the Nomenclature in AppendixB.
Taking the density as an example, we write the decomposition as
ρ = ρ1+ ρ2 (.)
where ρ1 is periodic. For the dri part ρ2 one can choose, for example, a linear function
ρ2= λρxx + λρzz (.)
where the coefficients λρx, λρz are the gradients of the dri part. ere is no
need to add an additive constant parameter to the dri part, since a constant additive term can be conceptually included in the periodic part. During a non-stationary simulation, the gradients vary with time.
