Validation of actuator disc circulation distribution for unsteady virtual blades model

A.N.Kusyumov , , S.A.Kusyumov , S.A.Mikhilov , G.N.Brkos

a Kazan National Research Technical University n.a. A N Tupolev, Russia

b Glasgow University, Glasgow, UK

Keywords: Surface circulation distribution Unsteady actuator disk model Blade tip vortices reproduction

ABSTRACT The actuator disc method is an engineering approach to reduce computer resources in computational fluid dynamics (CFD) simulations of helicopter rotors or aeroplane propellers.Implementation of an ac- tuator disc based on rotor circulation distribution allows for approximations to be made while reproduc- ing the blade tip vortices.Radial circulation distributions can be formulated according to the nonuniform Heyson-Katzoff“typical load”in hover.In forward flight, the nonuniform disk models include “azimuthal”sin and cos terms to reproduce the blade cyclic motion.The azimuthal circulation distribution for a for- ward flight mode corresponds to trimmed conditions for the disk rolling and pitching moments.The amplitude of the cos harmonic is analysed and compared here with presented in references data and CFD simulations results.

One of the primary requirements in helicopter design is the es- timation of the rotor effect on the helicopter fuselage.In particu- lar, the fuselage surface pressure distribution can be used as ini- tial data for structural analyses.The actuator disc (AD) method is a mathematical approach in computational fluid dynamics (CFD) to approximate a helicopter rotor or an airplane propeller.For this purpose, the Navier-Stokes or Reynolds averaged Navier-Stokes (RANS) equations are modified with a source terms distribution in the momentum equations.The “intensity”of the source terms is determined in the form of a “pressure jump”across the AD surface in accordance to the rotor thrust force.Since no surface is needed in the grid, the position of the sources reproducing the blades lo- cation can be changed without transformation of the initial grid.

In some CFD solvers (commercial ANSYS fluent CFD code, for example) the AD conception is realized in the form of a special type of boundary condition applied on the infinitely thin AD sur- face, localized at (and instead of) the fully resolved rotor disk.

Classical helicopter AD models provide a steady-state formula- tion of the RANS equations to estimate a time-averaged action of the rotor downwash on the helicopter fuselage and its surface ele- ments.Within the classical AD models, the source terms are con- tinuously distributed on the AD surface.

A more realistic approach for simulation of the unsteady space structure of the helicopter rotor wake is based on a variable and space localized source distribution similar to the fully resolved rotor blades planform.Such methods require solution of the un- steady Reynolds averaged Navier-Stokes (URANS) equations and are called as the actuator-blade methods.

For example, with the actuator-lines method [1,2] the rotor blades action is simulated by forces acting along filaments sources lines.Body forces are typically derived from the blade element mo- mentum (BEM) method.This classic BEM method is based requires corrections particularly near the blade tip region [3] .In Lynch et al.[4] it is noted that such correction can be applied using Prandtl’s approach [5] .

An alternative approach which reproduces the flow structure is the unsteady AD method: the disk surface is divided in the azimuthal direction with a time-varied pressure jump across the AD surface elements [ 4 , 6 ].In references [7–9] , such approach was termed unsteady virtual blades actuator (VBA) model.In Ref.[8] the AD surface was divided in 4 sectors according to the rotor blades number (Fig.1).

In forward flight the nonuniform AD models include “az- imuthal”terms reproducing the blade cyclic.The amplitudes of the azimuthal terms depend on the AD models and should be adjusted to provide trimmed flight disk loading.Reference [9] presents the AD mathematical formulations and comparison of several AD mod- els, based on prescribed AD circulation distributions.The AD1 model presented in Ref.[9] takes into consideration only the az- imuthal variation of the rotor circulation distribution.For the AD2 model the AD loading is determined with the Blade Element The- ory.A more complicated model AD3 is based on the “typical”cir- culation distribution [10] .The formulation of that model takes into account both azimuthal and radial circulation distributions and is described in detail in Ref.[ 11 ].

Another (different from Ref.[11]) approach, termed AD4, that also accounts for the radial and azimuthal circulation distribution, is presented in Ref, [9] .The AD loadings for different models are compared in Ref.[9] by the rotor surface normal force distribution obtained with numerical simulation for the PSP rotor [12] .

It should be noted that two kinds of “trimming coefficients”, determining the amplitude of the blade cyclic motion are offered [9] .A difference between the trimming coefficients is determined by a functional dependence (linear or nonlinear) on the rotor ad- vance ratio.The goal of this paper is the comparative analysis of the AD loading for the AD4 model with the linear and nonlinear forms of the trimming coefficients.For prescribed flight conditions the circulation distribution of the AD4 model is compared to the circulation distribution obtained with vortex theory [13] .In addi- tion the AD surface pressure distribution is compared to PSP rotor CFD simulation results (unlike Ref.[9] , which considered a com- parison with the surface normal force coefficient).

A widely accepted AD model, expresses the AD loading of a for- ward flying rotor as a function of the disk radiusrand azimuth angleΨ:

where the functions sin(r,Ψ)and cos(r,Ψ)determine the sin and cos pressure oscillations, and the coefficientsΔp0,Δp1s , andΔp2s depend on the rotor geometry and flight conditions.Using the ro- tor circulation functionΓ(r,Ψ)the local loading can be written as

where

hereρis the air density,Ωis the blade rotational speed,Ris the rotor disk radius, andμis the rotor advance ratio:

whereVtip=ΩRis the blade tip speed,Vis the forward rotor ve- locity, andαris the rotor disk plane tilt angle (positive for forward tilt).

The simplified circulation distribution (AD1 model) can be writ- ten as [ 9,13 ]:

while

whereCTis the rotor trust coefficient determined by the rotor thrustT:

A more advanced model is based on a "typical" circulation dis- tribution [10] .According to Ref.[9] the circulation distribution also depends on the normalized radial coordinate=r/R, and can be approximated as:

here

The azimuthal variation of the dimensionless disk circulation distributionis determined by the rotor advance ratioμand does not depend on the rotor thrust coefficient that is ac- counted by theΓ4term.The dimensionless disk circulation dis- tributionfor the hover mode is determined as [10] :

Two kinds of trimming coefficientsK(μ)andW(μ)can be de- termined for the sin and cos harmonic components [9] as linear and nonlinear functions of the advance ratio:

Substitution of Eqs.(12) and (13) into Eqs.(10) and (11) al- lows determination two kinds of functions for the dimensionless A D surface circulation distribution:and

One should note that both kinds trimming coefficients deter- mine the AD loading distribution are comparable, in general, to the CFD results.The radius-averaged azimuthal circulation distri- bution can be considered for clarification of the trimming coeffi- cients choice.One can determine the averaged 1D function

to compare with the simplest1(Ψ)dimensionless AD circulation.Substituting Eqs.(10) and (11) into Eqs.(14) and (15) gives a gen- eral expression

Substituting Eqs.(12) and (13) into Eq.(15) gives respectively

Figure 2 shows the function1(Ψ)in comparison to the aver- aged 1D functions41a(Ψ), and42a(Ψ)distribution for the rotor advance ratioμof 0.15.

From Fig.2, it follows that the andandfunction have two extremums, unlike thefunction, which has four extremums and one can expect that the42a(Ψ)function allows for better reproduction of real main rotor disk properties.

In Fig.3 , the dimensionless circulation distributionsandare shown forμ= 0.1 .Figure 3 shows a discrepancy between the dimensionlessandcirculation dis- tributions at the retreating blade area.Due to the different sine co- sine harmonic circulation components the AD model with thecirculation distribution is slightly higher in comparison with thecirculation near ≈270 °.

To validate the obtained disk load distributions one can analyze the rotor AD surface load distribution determined by theandfunctions in comparison with data presented in ref- erences.The theoretical circulation distribution obtained from the vortex theory is presented in Ref.[13] at the main rotor disk sec- tion= 0.7 for the rotor parameters: the rotor solidityσ= 0.07 , the rotor trustCTcoefficient of 0.012 and the advancing ratio of 0.15.

Figure 4 presents the normalized41(,Ψ)and42(,Ψ)func- tions compared with a theoretical circulation distribution.The rel- ative theoretical circulation distribution is presented in Ref.[13] in the form

where the function(= 0.7)is determined by Eq.(9) for= 0.7 .

Figure 4 shows, that the normalized42(= 0.7,Ψ)distribu- tion agrees very well with the vortex theory results.The trimming coefficients determined by expressions Eq.(13) (nonlinear form) thus provide better agreement of the rotor disk circulation distri- bution in comparison with the coefficients of Eq.(12) (linear de- pendence on the advance ratio).

The theoretical AD loading can be compared with CFD results.For CFD simulation the four-bladed PSP [12] rotor was considered (details of the rotor geometry, simulation conditions and brief de- scription of a CFD solver are presented).In Ref.[9] the theoreti- cal and CFD normal force coefficient distributions for the PSP ro- tor were also compared.In Fig.5 a comparison of the AD load- ings are considered for the surface pressure jump across the disk (whichisused as boundary condition in VBA models).The dimen- sionless pressure jump distribution on the rotor disk surface is shown forCT= 0.016 ,μ= 0.35 .In Fig.5 a the dimensionless pres- sure jumpCFD(,Ψ)is determined by the normalized rotor disk load

hereMtipis the blade tip Mach number, andCnis the normal force coefficient.Figure 5 b shows the obtained distribution

whereΔp(r,Ψ)is determined by the expression (2) for thecirculation distribution.

Comparison of the CFD simulation results to the obtained forpressure jump distribution shows a satisfactory agree- ment.The VBA model approach is based on "typical" [10] circula- tion distributions on the rotor disk and does not take into account the specific blade design or any particular rotor trimming method.For this reason, the considered AD model shows some discrepancy of the rotor disk load near the azimuth angleΨof 90 °compared to the CFD data.Nevertheless, the AD model predicted the high disk load forΨ≈45 °and 135 °at the disk radius 0.75Rand the lower values of the disk load near the rotor root part forΨ≈45 °.

Circulation distribution on the surface nonuniformly loaded ac- tuator disk model is analyzed developed using the “typical law”of the helicopter main rotor disk circulation distribution.The actuator disk model contains “trimming coefficients”, which determine disk circulation distribution taking into account the sin and cos circu- lation components.Circulation distribution on the disk surface is analyzed for two kinds of the trimming coefficients, which linear or nonlinear depend on the rotor advance ratio.

Both linear and nonlinear models yield the assigned thrust co- efficient value and satisfied the trimming conditions of the AD load for forward flight.However, the nonlinear AD model better agrees with the vortex theory prediction and with the rotor CFD simula- tion results.Comparison to the vortex theory results at the 75% of the rotor radius shows good agreement for the azimuthal circula- tion distribution, including peak to peak values and their location.Comparison to the CFD simulation results for the four-bladed rotor shows that the AD model predicted well the disk load for different azimuth angles and rotor disk radius, excluding the azimuth area near 90 °.

Declaration of Interest Statement

We confirm that the manuscript has been read and approved by all named authors and that there are no other persons who satis- fied the criteria for authorship but are not listed.We further con- firm that the order of authors listed in the manuscript has been approved by all of us.

Declaration of Competing Interest

The authors declare no conflict of interest.

Acknowledgement

Work of Russian coauthors was supported by the grant " FZSU- 2020-0021 " (No.075-03-2020-051/3 from 09.06.2020) of the Min- istry of Education and Science of the Russian Federation.