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Research article Special Issues

Reinforcement learning-based optimization of locomotion controller using multiple coupled CPG oscillators for elongated undulating fin propulsion

  • Received: 10 August 2021 Accepted: 08 November 2021 Published: 19 November 2021
  • This article proposes a locomotion controller inspired by black Knifefish for undulating elongated fin robot. The proposed controller is built by a modified CPG network using sixteen coupled Hopf oscillators with the feedback of the angle of each fin-ray. The convergence rate of the modified CPG network is optimized by a reinforcement learning algorithm. By employing the proposed controller, the undulating elongated fin robot can realize swimming pattern transformations naturally. Additionally, the proposed controller enables the configuration of the swimming pattern parameters known as the amplitude envelope, the oscillatory frequency to perform various swimming patterns. The implementation processing of the reinforcement learning-based optimization is discussed. The simulation and experimental results show the capability and effectiveness of the proposed controller through the performance of several swimming patterns in the varying oscillatory frequency and the amplitude envelope of each fin-ray.

    Citation: Van Dong Nguyen, Dinh Quoc Vo, Van Tu Duong, Huy Hung Nguyen, Tan Tien Nguyen. Reinforcement learning-based optimization of locomotion controller using multiple coupled CPG oscillators for elongated undulating fin propulsion[J]. Mathematical Biosciences and Engineering, 2022, 19(1): 738-758. doi: 10.3934/mbe.2022033

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  • This article proposes a locomotion controller inspired by black Knifefish for undulating elongated fin robot. The proposed controller is built by a modified CPG network using sixteen coupled Hopf oscillators with the feedback of the angle of each fin-ray. The convergence rate of the modified CPG network is optimized by a reinforcement learning algorithm. By employing the proposed controller, the undulating elongated fin robot can realize swimming pattern transformations naturally. Additionally, the proposed controller enables the configuration of the swimming pattern parameters known as the amplitude envelope, the oscillatory frequency to perform various swimming patterns. The implementation processing of the reinforcement learning-based optimization is discussed. The simulation and experimental results show the capability and effectiveness of the proposed controller through the performance of several swimming patterns in the varying oscillatory frequency and the amplitude envelope of each fin-ray.



    Aeroengine is an important equipment to propel the aircraft forward and its total thrust is the sum of the thrust generated by the core engine and the turbine fan. The air flow state between adjacent blades of the turbine fan determines the thrust value of aeroengine. Since the experiments cost a lot, numerical simulations become an essential part of the blade geometry design and optimization [5,10,2,12,7]. Nevertheless, many difficulties arise in numerical simulation such as nonlinearity, high Reynolds number, complex three-dimensional (3D) geometrical domains, and boundary layer effect (see, e.g., [9]). In order to alleviate these difficulties, we will use the dimension splitting method to simulate the airflow state of aeroengine blade fan.

    Based on the article [6], we establish a semi-geodesic coordinate system (called R-coordinate system), whose two basis vectors are on the manifold, and the other is along the hub circle. Thus, Navier-Stokes equations (NSEs) in the R-coordinate system can be rewritten as a set of membrane operator equations on the blade surface, and the bending operator equations along the hub circle [8]. By using Euler central difference scheme to approximate the third variable, the 3D NSEs become a series of two-dimensional (2D) equations with three variables. After successively iterations, the approximate solution to the NSEs can be obtained. Obviously, the significant feature of this new method is this method only solves the 2D problem in each sub-domain. In addition, it can alleviate the boundary layer effect by approaching adjacent surfaces, and the parameterized surface provides convenience for blade design and optimization.

    The purpose of this work is to introduce our proposed method to simulate the flow state of the channel of aeroengine turbine fan. A lot of work has been carried out in the field of viscous flow and its applications in aeroengine turbine [16,15,4,11,14,13,3]. In this paper, a toy model is designed to give a comparison between our novel method and traditional methods. It turns out the new method shows a good performance for the toy model. Then we apply the proposed method to simulate the flow state of aeroengine turbine fan.

    The present paper is built up as follows. In Section 2, some essential differential geometry knowledge is briefly introduced, then the R-coordinate system is established. Meanwhile, the NSEs' new form, splitting method and the variational form are formulated in Sections 3. Furthermore, we derive the finite element form in Section 4. Section 5 presents the numerical results, which contain the comparison of the new method and traditional methods, and the simulation results of aeroengine turbine fan.

    In this section, we establish a new coordinate system according to the geometric shape of the blade and give the relationship between the new coordinate system and the rectangular coordinate system and the cylindrical coordinate system. To express concisely and clearly, we let Greek letters α,β, and Latin letters i,j, range over the values {1,2} and {1,2,3}, respectively. Einstein summation convention is adopted in tensor analysis in the sequel.

    Compared with the size of the aircraft engine fan, the thickness of the fan blade can be neglected. Thus the blade surface is considered as a 2D surface in R3 in this article, which is a connected subset D defined in R2 and mapped to the range (D) by the injective map into R3. Suppose is smooth enough, any point x=(r,z)¯D in the Gaussian coordinate system on the surface (x) can be expressed as in the cylindrical coordinate system

    (x)=rer+rΘ(r,z)eθ+zk. (2.1)

    Here ΘC2(D) is a smooth function of radian, (er,eθ,k) are basis vectors of cylindrical coordinate system on the fan.

    The channel Ωε is determined by the boundary Ωε=ΓinΓoutΓtΓb+, where Γin,Γout are inlet and outlet, Γt,Γb are top surface (shroud) and bottom surface (hub), and +, are positive pressure surface and negative pressure surface (see Figure 1 and Figure 2). Denote Nb as the number of blades and ε=π/Nb, then rotating one blade 2ε can get another which means there exists a family of single-parameter surfaces ξ covering the flow channel by mapping (x;ξ): Dξ, that is

    Figure 1.  Aeroengine turbine fan.
    Figure 2.  Channel between two adjacent blades.
    (x;ξ)=rer+rθeθ+zk, (2.2)

    where θ=εξ+Θ(r,z) is the rotation angle.

    Let

    x1=z,x2=r,ξ=ε1(θΘ(x)),

    it is clear that the Jacobian matrix J((r,θ,z)(x1,x2,ξ))=ε is nonsingular. Thus we establish a new curvilinear coordinate system (called R-coordinate system) (x1,x2,ξ)

    (r,θ,z)(x1,x2,ξ):x1=z,x2=r,ξ=ε1(θΘ(x)). (2.3)

    In the R-coordinate system, the fixed region Ω={(x1,x2,ξ)|(x1,x2)D,1ξ1} is mapped to a channel Ωε={(x1,x2,ξ)=x2er+x2(εξ+Θ(x1,x2))eθ+x1k, (x1,x2,ξ)Ω}.

    Since the new coordinate is established, these basis vectors become a bridge to communicate different coordinate systems. Let (x,y,z) and (r,θ,z) denote Cartesian coordinate system and cylindrical coordinate system, respectively, which are fixed on impeller with angular velocity ω in a 3D Euclidean space. Let (i,j,k) and (er,eθ,k) be the basis vectors of (x,y,z) and (r,θ,z), respectively, which satisfy

    {er=cosθi+sinθj,eθ=sinθi+cosθj,i=cosθersinθeθ,j=sinθer+cosθeθ.

    The basis vectors of the R-coordinate system are denoted as (e1,e2,e3), which are defined as

    {eα=α=αxi+αyj+αzk,α=1,2,e3=ξ()=xξi+yξj+zξk, (2.4)

    where

    {x:=x(x1,x2,ξ)=rcosθ=x2cos(εξ+Θ(x1,x2)),y:=y(x1,x2,ξ)=rsinθ=x2sin(εξ+Θ(x1,x2)),z:=z(x1,x2,ξ)=x1.

    By straightforward calculation, the relationship between base vectors is as follows

    {e1=x2Θ1eθ+k=x2sinθΘ1i+x2cosθΘ1j+k,e2=Θ2x2eθ+er=(cosθx2sinθΘ2)i+(sinθ+x2cosθΘ2)j,e3=x2εeθ=εx2sinθi+εx2cosθj,er=e2ε1Θ2e3,eθ=(εx2)1e3,k=e1ε1Θ1e3,i=cosθe2(ε1cosθΘ2+(εx2)1sinθ)e3,j=sinθe2+((εx2)1cosθε1Θ2sinθ)e3, (2.5)

    where Θα=Θxα.

    The metric tensor aαβ of the surface ξ is defined as

    aαβ:=eαeβ=δαβ+r2ΘαΘβ,

    and it is easy to see that aαβ is nonsingular and independent of ξ as follows, that is

    a=det(aαβ)=1+r2|Θ|2>0, (2.6)

    where |Θ|2=Θ21+Θ22.

    Similarly, the covariant and contravariant components of metric tensor are gij and gij, which are defined as

    gij=eiej,gij=eiej.

    By calculation, they can be expressed as (cf.[5])

    {gαβ=aαβ,g3β=gβ3=εr2Θβ,g33=ε2r2,gαβ=δαβ,g3β=gβ3=ε1Θβ,g33=(εr)2a. (2.7)

    As described in our previous article [6], for given ξ, both surfaces ξ and have the same geometric characteristics and interested readers can refer to the article [6].

    In this section, we derive the flows state governed by the incompressible rotational NSEs through employing differential operators in the R-coordinate system. Because of the lighter mass of air, its body force is neglected and the rotational NSEs can be expressed as

    {utνΔu+(u)u+2ω×u+p=ω×(ω×r),divu=0. (3.1)

    For a time-dependent problem, we usually discretize the problem in time and solve the static problem at each time step, then the time-dependent problem is reduced to a static problem at each time step. If we consider an implicit time discretization (e.g., the backward Euler method) to equation (3.1) with time-step size κ, and multiply κ to it, then we have a static problem

    {uνκΔu+κ(u)u+2κω×u+κp=κω×(ω×r)+fu,divu=0, (3.2)

    where fu is the velocity value of previous time step.

    Assume that Γijk and juk are the Christoffel symbol and the covariant derivative in space, while Γαβσ and βuα are the Christoffel symbol and the covariant derivative on surface, their expression can be obtained by

    Γijk=eiejk,kui=uixk+Γikmum,Γαβσ=eαeβλ,βuα=uαxβ+Γαβσuσ,

    where eij=iej is the first-order partial derivative of basis vectors.

    Lemma 3.1. [6] By denoting (eα,e3) as the basis vectors in the coordinate system (x1,x2,ξ), the parts of NSEs in the new coordinate system have the following conclusions.

    1). The Laplace operator Δu:=gijiju can be written as

    Δui=˜Δui+2g3γγuiξ+g332uiξ2+Li3γuγξ+Li33u3ξ+Liσγσuγ+Li0γuγ+δi3L3σ3σu3, (3.3)

    where

    {Lα3γ=(εr)1(Θ2δαγ+2aδ2αΘγ)˜ΔΘδαγ,L33γ=2(εr)2(r2ΘβΘβγr1δ2γ),Lα33=2rδ2α,L333=(εr)1(r˜ΔΘ+Θ2),Lασγ=2rδ2αΘσΘγ(r|Θ|2r1)δ2σδαγ,L3σγ=2(εr)1(δ2σΘγ+rΘγσ)Lα0γ={Θγ[(a+1)Θ2+r˜ΔΘ]r2δ2γ+rΘβΘβγ}δ2α,L30γ=(εr)1(3Θ2γ+2(r2Θ2αΘα+rΘ22)Θγ+)rΘααγ+r2Θ2δ2γ,L3σ3=(3r1r|Θ|2)δ2σ, (3.4)

    and ˜ΔΘ=δαβΘαβ,˜Δuα=δλσλσuα.

    2). Its related items coriolis force and centrifugal force satisfy, respectively,

    C=2κω×u=κCijujei, (3.5)
    fic=κεijkωj(ω×r)k=κεijkgjmωmεklnωlrn=κεijkgjmωmεklnωlgn2r, (3.6)

    where C1j=0,C2α=2ωrΘα,C23=2εωr,C3α=2ω(rε)1a2α,C33=2ωrΘ2.

    3). The pressure gradient is p=gijpxj, which can be represented as

    p=[gαβpxβ+gα3pξg3βpxβ+g33pξ]=[pxαε1Θαpξε1Θβpxβ+(rε)2apξ].

    4). The nonlinear term in equation (3.1) is

    B(u,u)=κ(u)u=κBi(u)ei=κ[uββuα+u3uαξ+nαkmukumuββu3+u3u3ξ+n3kmukum], (3.7)

    where

    {nαλσ=0,nα3β=nαβ3=rεδ2αΘβ,nα33=rε2δ2α,n333=rεΘ2,n3λσ=(rε)1(a2λΘσ+δ2σΘλ)+ε1Θλσ,n33β=n3β3=r1a2β.

    5). The mass conservation formula can be rewritten as

    div u=uαxα+u3ξ+u2r,divu=uαxαrΘ2Θσuσ. (3.8)

    By Lemma 3.1, the incompressible rotational NSEs in the R-coordinate system can be rewritten as

    {ui+κ{uββui+u3uiξ+nikmukum+Cijujν[˜Δui+2g3γγuiξ+g332uiξ2+Li3γuγξ+Li33u3ξ+Liσγσuγ+Li0γuγ+δi3L3σ3σu3]+giβpxβ+gi3pξ}=Fi,uβxβ+u3ξ+u2r=0, (3.9)

    where

    Fi=fic+fiu.

    And its channel region and boundaries are

    {Ω={(x1,x2,ξ)|(x1,x2)D,1ξ1,},Ω=ΓinΓout+ΓtΓb. (3.10)

    The initial and boundary value conditions are

    u|t=0=uo,u|+=0,u|Γin=uin. (3.11)

    In this section, the finite-element-difference method will be presented. The first step is to divide the interval [1,+1] into N-subintervals with step size τ=2N, that is

    [1,+1]=N1k=0[ξk,ξk+1],ξk=1+kτ,k=0,1,N1.

    Then the domain Ω is split into N-layers Ω=N1k=0{D×{ξk,ξk+1}} and the central difference is used to replace the derivative with respect to the variable ξ, i.e.,

    wξwk+1wk12τ,2wξ2wk+12wk+wk1τ2. (3.12)

    where wk:=w(x,ξk). In order to simplify the equation, we denote that

    [w]+k:=wk+1+wk1τ2,[w]k:=wk+1wk12τ. (3.13)

    Plugging (3.13) into (3.9), we obtain

    {ui+κ{uββui+u3[ui]k+nikmukumν[˜Δui+2g3γγ[ui]k+g33([ui]+k2uiτ2)+Li3γ[uγ]k+Li33[u3]k+Liσγσuγ+Li0γuγ+δi3L3σ3σu3]+giβpxβ+gi3[p]k+Cijuj}=˜Fik,uαxα+[u3]k+u2r=0, (3.14)

    where

    ˜Fik:=1τξk+1ξkFi. (3.15)

    And its boundary conditions are

    uk|γ0=0,uk|γin=uin,D=γ0γinγout, (3.16)

    where γ0=(ΓtΓb){ξ=ξk},γin=Γin{ξ=ξk},γout=Γout{ξ=ξk}.

    Introduce the Hilbert space V(D) by

    V(D)={uH1(D)×H1(D)×H1(D),u=0|γ0γin},

    then its inner product and norms are given, respectively, by

    (w,v)D=Ωaijwivjadx,aij={aαβ=aαβ,aα3=a3α=0,a33=1},|w|21,D=αjαwj20,D,w20,D=jwj20,D,w21,D=|w|21,D+w20,D.

    Without the ambiguity, the index `D' is often omitted.

    For clarity and simplicity, we denote

    {Li(k):=κν(˜Δuik+Liσγσuγk+Li0γuγk2g33τ2uik+δi3L3σ3σu3k)+uik,Bi(k):=κ(uβkβuik+nilmulkumk),Ci(k):=κCimumk,Si:=κν(2g3γγ[ui]k+g33[ui]+k+Li3γ[uγ]k+Li33[u3]k)+κu3k[ui]k,Pi:=κgi3[p]k. (3.17)

    Meanwhile, we denote ˆFik=˜FikSiPi, then the 2D-3C NSEs on manifold ξ can be written as

    {Lα(k)+Bα(k)+Cα(k)+κgαβpkxβ=ˆFαk,L3(k)+B3(k)+C3(k)+κg3βpkxβ=ˆF3k,uαxα+[u3]k+u2r=0. (3.18)

    Thus, the variational problem corresponding to the boundary value problem of the NSEs is as follows:

    {Seek ukL(0,T;V(D))+uin, pkL2(D),k=0,1,2,,N1,s.t.a(uk,v)+(C(uk),v)+(L(uk),v)+b(uk,uk,v)(pk,m(v))=(ˆFk,v)<h,v>γout,vV(D),(uαkxα+u2kr,q)=([u3]k,q),qL2(D), (3.19)

    where the linear, bilinear and trilinear forms are given, respectively,

    {a(uk,v)=νκ(aijλuik,λvj)+νκ(ˆaijuik,vj)+(aijui,vj),ˆaij=aijr2aα2τ,(C(uk),v)=κ(Ci,vi),Cβ=aαβCαmumk,C3=C3mumk,(L(uk),v)=νκ(Li,vi),Lβ=λaαβλuαkaαβ(Lασνσuνk+Lα0νuνk),L3=(L3σλσuλk+L3σ3σu3k+L30σuσk),b(uk,uk,v)=(κaijBi(k),vj),(ˆFk,v)=(aijˆFik,vj),<h,v>γout=γoutκμaijλuikvjnλdl+γoutκpk(aαλvαεΘβv3)nλdl,m(v)=κ(αvαεΘββv3ε˜ΔΘv3). (3.20)

    In this section, we apply the Taylor-Hood elements, i.e., (P2,P1) Lagrange finite elements for the pair (u,p). Let Vh and Mh be the finite element subspaces corresponding to space V(D) and L2(D), respectively, which are

    Vh:={vhC0(ˉΩ);vh|KP2(K),KTh},Mh:={phC0(ˉΩ);ph|KP1(K),KTh}. (4.1)

    The product space Yh=Vh×Mh is subspace of Y=V(D)×L2(D) obviously. Then the variational problem (3.19) approximated by the standard Galerkin finite element method is

    {Seek whVh, phMh s.t.a(wh,vh)+(C(wh),vh)+(L(wh),vh)+b(wh,wh,vh)(ph,m(vh))=(ˆFk,vh)<h,vh>γout,vhVh,(wαhxα+w2hr,qh)=([w3h]k,qh),qhMh. (4.2)

    Suppose the finite element basis functions are denoted as

    φi(x),i=1,2,,NG1, ϕi(x), i=1,2,NG2,

    where NG1 and NG2 are the total number of nodes, respectively. Finite element expansion of wh,ph are

    wmh=NG1i=1Ximφi(x),ph=NG2i=1Piϕi(x),m=1,2,3,vkh=NG1i=1Yikφi(x),qh=NG2i=1Qiϕi(x),k=1,2,3. (4.3)

    Assume solution vector Wm and test vector Vk are

    Wm={X1m,X2m,XNG1m}T,W={W1,W2,W3}T,Vk={Y1k,Y2k,YNG1k}T,V={V1,V2,V3}T,P={P1,,PNG2}T,Q={Q1,,QNG2}T. (4.4)

    Substitute (4.4) into (4.2), we obtain the result on 2D membrane operator:

    {a(wh,vh)=KαβijXiαYjβ,(L(wh),vh)=LτβijXiτYjβ,(C(wh,ω),vh)=CτβijXiτYjβ+C3βijXi3Yjβ,A0(wh,vh)=a(wh,vh)+(L(wh),vh)+(C(wh,ω),vh)=[KτβijXiτ+K3βijXi3]Yjβ,b(wh,wh,vh)=blm,βik,jXilXkmYjβ,(ph,m(vh))=BβijPiYjβ (4.5)

    where

    {Kαβij=κν[(aαβλφi,λφj)+α2τ(aαβar1φi,r1φj)]+(aαβφi,φj),Lτβij=κν(λaαβλφiδατ,φj)κνaαβ((Lασνσδντ+LασνΓνσλδλτ+Lα0νδντ)φi,φj),Cτβij=(κaαβCατφi,φj),C3βij=(κaαβCα3φi,φj),Kτβij=Kαβijδτα+Lτβij+Cτβij,K3βij=C3βij,blm,βik,j=κ(aαβ{δlλφi[λφkδαm+Γαλσφkδσm]+nαlmφiφk},φj),Bβij=κ(ϕi,βφj).

    Proof. In this proof, we only prove (L(wh),vh). Noting that

    σwν=σwν+Γνσλwλ,σw3=σw3,

    we obtain

    (L(wh),vh)=κ(νaαβLασνσwνhνaαβLα0νwνh,vβh)+κ(νλaαβλwαh,vβh)=κ(νaαβLασν[σwνh+Γνσλwλh]νaαβLα0νwνh,vβh)+κ(νλaαβλwαh,vβh)=κν(λaαβλwαhaαβLασνσwνh,vβh)κν(aαβLασνΓνσλwλh+aαβLα0νwνh,vβh)=κν(λaαβλφiδατaαβLασνσφiδντ,φj)XiτYjβκν(aαβLασνΓνσλφiδλτ+aαβLα0νφiδντ,,φj)XiτYjβ=LτβijXiτYjβ.

    The remainder of the argument is analogous and is left to the reader.

    Next, we will give the discrete scheme on the right side of the formula (4.2). According to the definition of inner product in space V(D), this discrete scheme can be written as

    {(ˆFk,v)=(aαβˆFαk,vβ)=ˆFβjYjβ,<h,v>γout=κνaαβλφiXiαφjYjβnλdl+κaαβϕiPiφjYjαnβdl=ˆHβjYjβ.

    Thus, we obtain

    KτβijXiτ+K3βijXi3+blm,βik,jXilXkmBβijPi=Fβj, (4.6)

    where Fβj=ˆFβjˆHβj. Similarly, the bending operator can be discreted as

    {a(w3h,v3h)=K33ijXi3Yj3,(L(w3h),v3h)=Lτ3ijXiτYj3+L33ijXi3Yj3,(C(w3h,ω),v3h)=C3αijXiαYj3+C33ijXi3Yj3,A0(w3h,v3h)=[Kα3ijXiα+K33ijXi3]Yj3,b(w3,w3,v3)=blm,3ik,jXilXkmYj3,(ph,m(vh))=B3ijPiYj3, (4.7)

    where

    {K33ij=κν(λφi,λφj)+κνα2τ(ar1φi,r1φj)+(φi,φj),Lτ3ij=(κν[L3σλ(σφiδλτ+Γλστφi)+L30τφi],φj),L33ij=(κνL3σ3σφi,φj),Cα3ij=(κC3αφi,φj),C33ij=(κC33φi,φj),K33ij=K33ij+L33ij+C33ij,Kα3ij=Lα3ij+Cα3ij,blm,3ik,j=κ(φiδβlδ3mβφk+n3lmφiφk,φj),B3ij=κ(ϕi,εΘββφj+ε˜ΔΘφj).

    And its right terms are

    {(ˆF3k,v3)=(ˆF3k,Yj3φj)=ˆF3jYj3,<h3,v3>γout=κνλφiXi3φjYj3nλdlκεΘβϕiPiφjYj3nλdl=ˆH3jYj3.

    It is easy to verify it, so its proof will not be given here. Then the bending operators becomes

    Kα3ijXiα+K33ijXi3+bml,3ik,jXimXklB3ijPi=F3j, (4.8)

    where F3j=ˆF3jˆH3j.

    Finally, we obtain the finite element algebraic equations of equation (3.19), i.e.,

    {KτβijXiτ+K3βijXi3+blm,βik,jXilXkmBβijPi=Fβj,Kα3ijXiα+K33ijXi3+bml,3ik,jXimXklB3ijPi=F3j,MijXiα=(φi[Xi3]k,ϕj), (4.9)

    where

    Mαij=(αφi+r1φiδα2).

    In this section, for certain examples, the results of the new algorithm program and the traditional algorithm program will be compared to verify the accuracy of the new algorithm program. And the new algorithm program shows good performance. Then, based on the new algorithm, the flow state of the gas in the fan channel of an aeroengine is given.

    In this part, a simple model is provided to give the comparison between results obtained by the dimension splitting method (DS method) and traditional 3D method (T3D method). In this example, we adopt stationary model and assume Θ=0 and θ=[7.5,7.5], see Figure 3. In this case, the central surface of both methods is completely coincident. Figure 4 presents one blade of this model and it is also the shape of the central surface. DS method is concerned with the solution of 2D surface. Figure 5(a) and Figure 5(b) show the mesh of central surface, respectively. By comparison, we can see that the mesh of Figure 5(a) is more regular than Figure 5(b). That is because Figure 5(a) is one of the solving planes while Figure 5(b) is the projection of a 3D mesh on the central plane.

    Figure 3.  The channel of impellers when Θ=0.
    Figure 4.  One of blade in R-coordinate system.
    Figure 5.  The mesh of central surface generated by different methods.

    51 2D-manifolds and 4987 elements per surface are used to partition the channel and for DS method while 254867 elements are used for T3D method. In this model, we assume vin=1m/s and ω=10 rad/s. Figure 6Figure 9 show the comparison of results of the velocity distribution on the central surface. And Figure 10 shows the comparison of results of the pressure distribution on the central surface. By comparison, we know that the results obtained by 3D method and DS method are almost identical.

    Figure 6.  The comparisons of velocity magnitude calculated by different methods.
    Figure 7.  The comparisons of velocity u1 calculated by different methods.
    Figure 8.  The comparisons of velocity u2 calculated by different methods.
    Figure 9.  The comparisons of velocity u3 calculated by different methods.
    Figure 10.  The comparisons of Pressure calculated by different methods.

    As mentioned above, the flow state determines the maximum thrust that aeroengine turbine fan can provide. Our main purpose is using the DS method to simulate the flow state of the aeroengine turbine fan. In addition to the alleviation of boundary layer effects and make parallelism easier, DS method can give one clear design objective Θ for blade shape. Because the flow state mainly depends on the shape of blades, Θ can be determined by the inverse problem method according to the required thrust value.

    In this article, we adopt the blades provided by the partner airlines which shape can be shown in Figure 15. Its meshes in the R-coordinate system are shown in Figure 1, and 51 2D-manifolds are used in this case. The velocity at inlet is 80 m/s and the rotating angular velocity is 80 rad/s. Figure 11Figure 12 show the pressure distribution on the blade surface. Because of the rotation, positive pressure surface and negative pressure surface have different pressure distribution. The pressure distribution of the positive pressure surface is higher than that of the negative pressure surface. Because the blade is cocked up at the bottom (see Figure 15), the pressure distribution near the bottom of the blade is obviously different from other places. Through formula transformation, we present the 3D model of blade pressure and velocity in Figure 13Figure 14.

    Figure 11.  The pressure distribution of positive pressure surface.
    Figure 12.  The pressure distribution of negative pressure surface.
    Figure 13.  The 3D model of pressure distribution of positive pressure surface.
    Figure 14.  The 3D model of pressure distribution of negative pressure surface.
    Figure 15.  The blade mesh in R-coordinate system.

    Meanwhile, we show the velocity distribution at outlet in Figure 16. The velocity on the outside is significantly higher than that on the inside, and the velocity in the middle of the channel is also higher than that near the blade, which is consistent with common sense. From Figure 16, we can find that the maximum velocity is located at the edge of the blade, which can reach 130 m/s.

    Figure 16.  Velocity distribution at outlet.

    In this work, we conduct numerical simulations for 3D flow in the flow channel of the aeroengine turbine fan based on the dimension splitting method. However, the simulation of the flow states between the blades is not the ultimate goal. Our ultimate goal is to use the simulation results to design or optimize the blades. Meanwhile, the parameterized blade surface provides convenience for blade design and optimization. Thus, it will be the focus of follow-up research to optimize the blade by combining the inverse problem. In addition, compressible flow and boundary layer phenomenons are also the focus of the follow-up research.

    The work of G. Ju was partially supported by the NSF of China (No. 11731006), the Shenzhen Sci-Tech Fund (No. JCYJ20170818153840322), and Guangdong Provincial Key Laboratory of Computational Science and Material Design No. 2019B030301001. The work of R. Chen was supported by the NSF of China (No. 61531166003) and Shenzhen Sci-Tech fund (No. JSGG20170824154458183 and ZDSYS201703031711426). The work of J. Li was partially supported by the NSF of China (No. 11971221) and Shenzhen Sci-Tech fund (No. JCYJ20190809150413261). The work of K. Li was supported by the NSF of China under the grant No. 10971165 and 10771167.



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