Investigation on influence mechanism of length-to-diameter ratio on fluid-structure interaction characteristic of serpentine nozzle

The effect of different length-to-diameter ratios on the fluid-structure coupling characteristics of double serpentine nozzle for turbofan engine was studied by using serial bidirectional loose coupling method. The results show that the structural deformation of the serpentine nozzle is mainly located at the downstream channel of the first bend of the nozzle and the upper wall of the exit of the nozzle, and with the increase of the length-to-diameter ratio, the deformation amount of the nozzle gradually increases. The maximum deformation of the upper and lower walls of the serpentine nozzle occurs in the downstream region of the first bend of the serpentine nozzle. The fluid-structure coupling effect has an influence on the flow field characteristics and aerodynamic performance of the serpentine nozzle. When the length-to-diameter ratio is 2.4, the influence of fluid-structure coupling effect is lowest, and the total pressure recovery coefficient decreases by 0.40%, the flow coefficient decreases by 5.34%, and the thrust coefficient decreases by 0.92%. When the length-to-diameter ratio is 3, the influence of fluid-structure coupling effect is highest, and at this time, the total pressure recovery coefficient decreases by 0.48%, the flow coefficient decreases by 5.95%, and the thrust coefficient decreases by 1.12%.


Introduction
In order to achieve higher levels of mission execution, combat survivability and multi-purpose functions in actual combat missions, the new generation of military aircraft must have superior stealth capabilities.Low observable serpentine nozzle is one of the key technologies to improve the stealth performance of the exhaust system of aero-engine.Along with the long-term high-speed flight of the aircraft, the serpentine nozzle will continue to work under the multi-physical field load environment composed of aerodynamic load, thermal load and structural load [1,2].The geometric configuration of the serpentine nozzle is more complex than that of the traditional axisymmetric nozzle.The special geometric configuration makes the nozzle more sensitive to the internal flow characteristics, more prominent in elastic characteristics, and the complex geometric configuration leads to its internal flow showing typical non-uniform characteristics, and the distribution of aerodynamic load on the nozzle wall is very complex [3,4].The drastic change of aerodynamic load makes the serpentine nozzle show complex deformation characteristics, and the deformation characteristics of the serpentine nozzle in turn affect The 17th Asian International Conference on Fluid Machinery (AICFM 17 2023) Journal of Physics: Conference Series 2707 (2024) 012107 IOP Publishing doi:10.1088/1742-6596/2707/1/012107 2 the flow characteristics of the serpentine nozzle, thus changing the distribution of aerodynamic load on the nozzle wall.They interact with each other, forming the fluid-structure coupling (also known as "aeroelasticity") effect of the serpentine nozzle [5,6], resulting in complex aeroelastic phenomena such as stress concentration and large structural deformation of the serpentine nozzle, significantly changing the aerodynamic characteristics of the nozzle, seriously affecting the stability and tactical performance of the serpentine nozzle, and bringing immeasurable damage to the combat aircraft.
Some researchers have carried out some studies on the structural deformation and aeroelasticity of serpentine nozzle under the multi-physical field load environment.Smith et al. [7] studied the aerodynamic deformation mechanism of the serpentine nozzle equipped with the "Eikon" stealth unmanned aerial vehicle based on the bidirectional loose coupling method, and introduced the deformation suppression method.Nikhil et al. [8] used ANSYS finite element software to carry out the unidirectional fluid-structure-thermal coupling characteristics study of the exhaust system composite structure, and compared the difference of the influence of pressure load and temperature load on the structural deformation characteristics of the exhaust system.Urbanczyk et al. [9] developed a multi-physical field analysis and design optimization software COMANDO, and used the unidirectional fluid-structure coupling method to optimize the wall thickness of supersonic binary nozzle under two different constraint conditions.Ma Xiangtao et al. [10] established a parallel multi-scale optimization algorithm framework, and applied it to the material distribution optimization of composite plate and shell structure, to achieve the maximum design of structural natural frequency under constraint conditions.Sun Peng et al. [11] numerically simulated the structural deformation characteristics and internal/external flow characteristics of double-annular serpentine convergent nozzle under fluid-structure coupling effect, and the results show that the deformation position of serpentine nozzle after experiencing fluid-structure coupling effect is mainly distributed on the downstream wall of the first bend and the nozzle outlet, resulting in a significant reduction of nozzle aerodynamic performance.Gu Rui et al. [12,13] carried out the fluid-structure coupling vibration characteristics study of single-side expansion nozzle at different geometric adjustment positions based on MpCCI coupling platform, and analyzed the fluid-structure coupling vibration characteristics of single-side expansion nozzle.The results show that the fluid-structure coupling effect has little influence on the thrust performance of this model, but reducing the thickness of the nozzle lip plate will increase the influence of fluid-structure coupling effect on the overall performance of the nozzle.
The geometric configuration of the serpentine nozzle is very complex, and there are many geometric parameters involved in the design process, which have a significant impact on the flow characteristics of the serpentine nozzle.In practical fighter applications, the length-to-diameter ratio determines the length of the lower plate of the low observable serpentine nozzle and its integrated configuration with the rear fuselage of the aircraft, and is one of the important parameters in the design process [14].Crowe et al. [2] studied the effect of key parameters on the outlet temperature distribution and aerodynamic performance of the serpentine convergent-divergent exhaust system.The study found that when the width-to-height ratio of the throat is increased for small length-to-diameter ratio serpentine nozzles, the average temperature on the upper surface of the nozzle decreases, the average temperature on the lower surface increases, and the flow coefficient of the serpentine nozzle increases significantly.Wang Ding et al. [15] designed a double serpentine binary exhaust nozzle with small middle section eccentricity ratio and outlet eccentricity ratio greater than zero, and numerically studied its flow characteristics and thrust characteristics.The results show that small middle section eccentricity ratio and large length-to-diameter ratio effectively reduce the thrust loss of serpentine nozzle, and the thrust of double serpentine binary exhaust nozzle increases by 0.2% compared with axisymmetric nozzle.Sang Xueyi et al. [16] carried out a study on the effect of length-to-diameter ratio and eccentricity ratio on the performance of double serpentine binary nozzle.The study found that with the increase of length-to-diameter ratio, the nozzle thrust increases continuously, and with the increase of eccentricity ratio, the nozzle thrust decreases continuously.Sun Xiaolin et al. [17] numerically studied the effect of length-to-diameter ratio on the aerodynamic performance and flow characteristics of single-annular double serpentine nozzle, and the results show that when the length-to-diameter ratio is too small, flow separation occurs inside the nozzle, resulting in a sharp decline in aerodynamic performance of the nozzle.When the pressure drop ratio is 2.4, as the length-to-diameter ratio changes from 3 to 2.2, the thrust coefficient decreases by 6.7%.It can be seen that this key geometric parameter has a significant influence on both aerodynamic performance and infrared radiation characteristics of serpentine nozzle.
In summary, current public literature has paid attention to structural response characteristics of serpentine nozzle under multi-physical field, while studies on influence of length-to-diameter ratio, a key geometric parameter, on serpentine nozzle mainly focus on aerodynamic and infrared radiation characteristics, and studies on fluid-structure coupling effect of double serpentine nozzle based on different length-to-diameter ratios under viscous flow have not been involved yet, and influence of length-to-diameter ratio on flow characteristics and aerodynamic performance of serpentine nozzle under fluid-structure coupling effect has not been analyzed in depth.Therefore, this paper carries out bidirectional fluid-structure coupling numerical study of double serpentine nozzle under different length-to-diameter ratios, analyzes influence of this key geometric parameter on coupling characteristics and flow characteristics of serpentine nozzle, and further summarizes influence rules of length-to-diameter ratio on its aerodynamic performance and structural characteristics, providing basis for design of double serpentine nozzle configuration that satisfies both structural strength reliability and good aerodynamic performance.

Computational model
The research object of this paper is a double-annular serpentine nozzle based on a certain type of mixed-flow turbofan engine, which consists of an exhaust mixer and a double serpentine convergent nozzle, as shown in Figure 1.The exhaust mixer comes from the end structure of the turbofan engine, and the outlet area of the serpentine nozzle is obtained by calculating the overall performance of the engine.The exhaust mixer includes tail cone, lobed mixer and inner/outer annular channel structure.The lobed mixer shows an annular expansion structure along the axial direction, and its surface is evenly distributed with 12 identical lobed configurations along the circumferential direction.The tail cone is located in the center of the inner annular channel and has a large axial size, and its profile end extends to the inside of the first bend channel of the serpentine nozzle.The serpentine nozzle achieves the improvement of infrared stealth capability by blocking the high-temperature components of the engine through profile, and this paper designs different configurations of serpentine nozzle based on the variable cross-section design method with multi-parameter coupling, through the design of nozzle centerline change rule, nozzle flow cross-section design along the path, and establishment of low observable design criteria.The three-dimensional centerline design of the serpentine nozzle is mainly based on the extension of the two-dimensional Lee curve method [18] in three-dimensional space, to meet the three-dimensional spatial attribute requirements of nozzle inlet/outlet position, and make the geometric configuration of the nozzle better layout in the limited space inside the fuselage.The establishment of low observable design criteria is achieved by completely blocking the high-temperature components of the engine by serpentine profile, so that it is impossible to detect high-temperature components at any detection angle, and reduce infrared radiation intensity.
Figure 2 shows a schematic diagram of key geometric parameters of S-bend section, where nozzle inlet area is determined by nozzle diameter nozzle diameter is determined by mixer outlet diameter, and nozzle outlet area is determined by engine performance parameters.The main design parameters of serpentine nozzle include nozzle diameter (D), nozzle length (L), first bend channel outlet area (A 1 ), first bend aspect ratio (W 1 /H 1 ), first bend channel axial length (L 1 ), first bend longitudinal offset (ΔY 1 /L 1 ), second bend channel axial length (L 2 ), second bend longitudinal offset (ΔY 2 /L 2 ), nozzle exit aspect ratio (W e /H e ), equal straight section length (L 3 ).L 1 and L 2 are dimensionless as second bend to first bend length ratio (L 2 /L 1 ), and L 3 is dimensionless as equal straight section length to diameter ratio (L 3 /D).The values of dimensionless parameters formed by key geometric parameter combination under design state are shown in Table 1.Equal length/diameter ratio-L3/D 0.309 Figure 3 shows the low observability design criterion of the serpentine nozzle for completely shielding the high-temperature components, that is, the common tangent line MN of the upper and lower longitudinal lines of the nozzle passes through the upper point C of the nozzle exit or the lower point B of the nozzle inlet.By shielding at the two bends (points M and N) of the nozzle, the high-temperature components are completely shielded.There is a mutual coupling and mutual constraint relationship between the parameters of the serpentine nozzle, especially the second bend offset directly affects the shielding effect of the high-temperature components.When other design parameters remain unchanged, the length-to-diameter ratio is only related to the second bend offset.When the nozzle length-to-diameter ratio changes, the second bend offset of the nozzle will decrease as the length-to-diameter ratio increases.

Numerical methods
The numerical simulation of fluid domain for serpentine nozzle adopts CFD (Computational Fluid Dynamics) software Fluent for numerical simulation.Pressure-based solver for three-dimensional unsteady compressible Reynolds-averaged Navier-Stokes (N-S) equation is adopted.Turbulence model chooses SST k-ω model.Spatial discretization adopts second-order upwind scheme.Working medium is ideal gas.Structure domain for serpentine nozzle adopts CSD (Computational Structural Dynamics) software Abaqus for calculation.Structural analysis of structure domain adopts finite element method.Based on Newmark-β direct integration method, structural dynamics equation is solved in time domain.Solution adopts dynamic implicit analysis step.
The calculation grid and boundary conditions of fluid domain for serpentine nozzle are shown in Figure 5.The calculation domain is composed of hybrid grid splicing, including nozzle domain and far field domain.Nozzle structure deformation leads to continuous movement of flow field boundary, so nozzle domain is set as dynamic domain and tetrahedral unstructured grid is divided.Through grid smoothing and reconstruction to adapt to constantly changing flow field space, dynamic grid technology [19] is used to adjust grid movement rule in boundary layer, so that boundary layer grid and wall surface remain relatively static.Far field domain is set as static domain and hexahedral structured grid is divided.Grid in this area does not deform or reconstruct.Near wall grid is encrypted to meet calculation requirements of SST k-ω turbulence model.After grid independence verification, grid used for calculation is 5.38 million.Nozzle structure domain adopts hexahedral grid division, and finite element model is shown in Figure 6, where grid element type is 8-node linear non-conforming mode element.After grid independence verification, number of grids used for calculation is 80664.In the process of coupling solution, ground working condition of a certain type of turbofan engine is selected for unsteady flow field calculation, and calculation does not involve mutual transfer of temperature load and heat flux density.Both inner/outer annulus inlets adopt pressure inlet boundary, inlet inflow along axial direction, inlet total temperature and total pressure values are calculated by aero-engine performance simulation model, it is uniformly distributed on inlet surface of inner/outer annulus channel, specific parameter values are shown in Table 2. Nozzle wall surface adopts no-slip adiabatic wall surface boundary, nozzle outlet back pressure P b =1atm.Far field static domain outlet is set as pressure outlet boundary, outlet static pressure P 0 =1atm, static temperature T 0 =300K, other boundaries of nozzle dynamic domain and far field static domain are all set as far field boundaries.
In structure field, serpentine nozzle is assumed as cantilever beam structure, inlet surface adopts fixed support constraint, nozzle wall thickness is 4mm.Nozzle structure material chooses GH706 high temperature alloy, material properties include elastic modulus, density and Poisson's ratio, specific parameter values are shown in Table 3.Based on Rayleigh damping assumption [20], larger structural damping is set to eliminate possible structural vibration of nozzle, gravity and heat flow exchange of nozzle structure are ignored.Table 2. Boundary parameters of double serpentine nozzle.

Coupling solution method
Considering the solution accuracy and computational resources, the serial bidirectional loose coupling algorithm [20] is adopted for the fluid-structure coupling numerical method.The bidirectional fluid-structure loose coupling method is based on the Client-Server model to execute the serial coupling mechanism in the time domain [21,22].The calculation process is shown in Figure 7.  Due to the problems of time step lag and incomplete energy conservation on coupling surface, the bidirectional fluid-structure loose coupling method based on serial mechanism is of first-order time accuracy [23,24].In addition, the time scale of structural response is larger than that of flow, which leads to inconsistency of optimal time step for two physical fields.Therefore, it is necessary to adopt smaller coupling calculation time step to ensure accuracy and stability of calculation.

Coupling time step verification
Coupling time step has an important influence on high-efficiency and high-accuracy fluid-structure coupling simulation.Based on grid independence analysis, three groups of different coupling time steps ∆t 1 =1ms, ∆t 2 =2ms and ∆t 3 =4ms [7] are selected for fluid-structure coupling numerical calculation.When nozzle structure deformation reaches stable state, maximum deformation amount of upper wall downstream of first bend of serpentine nozzle and error value under different time steps are shown in Table 4.It is calculated that error of deformation amount of upper wall downstream of first bend of nozzle calculated by time steps ∆t 2 =2ms and ∆t 3 =4ms is 1.84%; while error of deformation amount of upper wall downstream of first bend of nozzle calculated by time steps ∆t 1 =1ms and ∆t 2 =2ms is only 0.77%.Therefore, adopting time step ∆t 2 =2ms can more accurately simulate fluid-structure coupling characteristics of serpentine nozzle on basis of speeding up calculation speed.

Coupling method verification
In order to verify numerical accuracy of serial bidirectional loose coupling algorithm and MpCCI coupling platform adopted in this paper, cold state experimental study of double serpentine nozzle scale-down model is carried out.In order to accurately describe deformation characteristics of serpentine nozzle, VIC-3D (Video Image Correlation-3D) non-contact full-field strain analysis measurement system is used to obtain deformation displacement distribution of key parts on nozzle wall surface.By spraying measurement speckle on wall surface and determining coordinate relationship between corresponding speckle image points in deformation before/after images, deformation shape displacement and strain cloud map of key parts on serpentine nozzle wall surface are obtained.Measurement accuracy of deformation displacement is 0.01mm and strain measurement accuracy is 0.005%.
Scale-down experimental model for serpentine nozzle is 10:1 scale-down model for double serpentine benchmark nozzle studied in this paper.SLA photosensitive resin is used as structure material.Wall thickness of experimental model is 1.6mm.Serpentine nozzle experimental model and its wall surface spraying measurement speckle effect are shown in Figure 8. Experimental boundary adopts ground working condition, outer annulus inlet pressure drop ratio is 1.74, inner annulus inlet pressure drop ratio is 1.68.5 shows the contour error of symmetry surface of serpentine nozzle model.Areas with large error between numerical prediction value and experimental measurement value are mainly located on upper wall surface of downstream channel of first bend and nozzle outlet position.Maximum relative error value at position of upper wall surface of downstream channel of first bend is 9.0%, maximum relative error value of upper wall surface at nozzle outlet is 8.8%, and maximum relative error value of lower wall surface at nozzle outlet is 9.1%.Compared with numerical/experimental errors of different types of nozzles at home and abroad [25,26] , where average error in literature [25] is 20%, minimum error in literature [26] is 7%, and error value between numerical simulation results and experimental measurement data in this paper is small, which is reasonable and reliable.In summary, serial bidirectional loose coupling algorithm adopted in this paper can accurately simulate flow field characteristics and structural deformation characteristics of double serpentine nozzle.

Nozzle deformation characteristics under different length-to-diameter ratios
In this paper, the influence of five different length-to-diameter ratio (L/R) parameters on the bidirectional fluid-structure coupling characteristics of serpentine nozzle is studied under ground working condition.The geometric configurations of serpentine nozzles with different length-to-diameter ratios are shown in Figure 11.The values of length-to-diameter ratio and the corresponding dimensionless longitudinal offset of the second bend are shown in Table 6.Under the design criterion of completely blocking high-temperature components, the dimensionless longitudinal offset of the second bend changes, and other geometric parameters are selected according to the research results in literature [14], which are the values under design state.The dimensionless longitudinal offset of the second bend reflects the bending curvature of the serpentine nozzle in the longitudinal direction.
When the length-to-diameter ratio increases, the longitudinal offset of the second bend gradually decreases, and the lateral expansion range of the nozzle profile increases, thus making the profile curvature of the serpentine nozzle along the longitudinal direction gradually decrease and along the lateral direction gradually increase.Therefore, as the length-to-diameter ratio increases, the entire serpentine nozzle geometric configuration becomes more gentle along the longitudinal bending channel and gradually increases along the lateral expansion angle.Due to the mutual influence of non-uniform aerodynamic load caused by complex geometric configuration of serpentine nozzle and complex deformation characteristics caused by elastic structure, fluid-structure coupling effect of serpentine nozzle is formed.When fluid-structure coupling effect of serpentine nozzle reaches dynamic balance, its structural deformation characteristics begin to stabilize.According to Figures 9 and 10, due to influence of serpentine bending configuration, structural deformation of serpentine nozzle is mainly distributed on upper/lower wall surfaces of serpentine nozzle.Due to no large curvature bending structure on side wall surface configuration, deformation amount of serpentine nozzle side wall surface is small.Among them, deformation of upper/lower wall surfaces of serpentine nozzle mainly shows wall bulge characteristic of downstream channel of first bend and upward bending deformation of nozzle equal straight section outlet along Y direction.
Figure 12 shows deformation amount distribution on upper wall surface of serpentine nozzle under coupling state with different length-to-diameter ratios.It can be seen from figure that deformation on upper wall surface of serpentine nozzle with different length-to-diameter ratios is mainly concentrated in downstream area of first bend and outlet area of equal straight section.For downstream area of first bend, as length-to-diameter ratio increases, deformation amount on upper wall surface in downstream area of first bend gradually increases.When length-to-diameter ratio is 2.2, local maximum value of deformation amount in downstream area is 19.1mm; when length-to-diameter ratio increases to 3, local maximum value of deformation amount in downstream area increases to 35.4mm.For outlet area of equal straight section, as length-to-diameter ratio increases, deformation amount at outlet gradually increases and reaches maximum when length-to-diameter ratio is 3.When length-to-diameter ratio is 2.2, local maximum value of deformation amount at outlet is 17.0mm; when length-to-diameter ratio increases to 3, local maximum value of deformation amount at outlet increases to 30.7mm.As length-to-diameter ratio increases, deformation distribution range on upper wall surface gradually expands, but distribution position has no significant difference.Figure 13 shows deformation amount distribution on lower wall surface of serpentine nozzle under coupling state with different length-to-diameter ratios.It can be seen from figure that deformation on lower wall surface of serpentine nozzle with different length-to-diameter ratios is mainly concentrated in downstream area of first bend, and there is no obvious deformation at outlet in lower wall surface area.As length-to-diameter ratio increases, deformation amount on lower wall surface in downstream area of first bend gradually increases.When length-to-diameter ratio is 2.2, local maximum value of deformation amount in downstream area is 20.1mm; when length-to-diameter ratio increases to 3, local maximum value of deformation amount in downstream area increases to 35.6mm.As length-to-diameter ratio increases, deformation distribution range on upper wall surface gradually expands, but distribution position has no significant difference.Combining Figures 12 and 13, it can be obtained that maximum deformation for upper/lower wall surfaces of serpentine nozzle within five kinds of length-to-diameter ratios all occurs in downstream area of first bend, distribution position is symmetrical along centerline, and change amount is approximately equal.Deformation on wall surface in downstream channel of first bend is mainly affected by circular-square configuration and rectangular cross-section shape in downstream channel of first bend for serpentine nozzle.As length-to-diameter ratio increases, shape for first bend channel gradually tends to "flat", channel side wall surface receives weaker aerodynamic load, channel upper/lower wall surface receives stronger aerodynamic load, and instability of serpentine nozzle first bend channel geometric configuration is enhanced.Therefore, deformation amount on wall surface of first bend gradually increases.Deformation on wall surface at outlet of equal straight section is mainly affected by bending configuration of serpentine nozzle.As length-to-diameter ratio increases, longitudinal offset of second bend gradually decreases, bending degree decreases, axial length of first and second serpentine channels increases, local aerodynamic load increases, which leads to increasing influence of bending configuration of second bend on nozzle deformation with increase of length-to-diameter ratio.In summary, as length-to-diameter ratio increases, instability of serpentine nozzle structure is significantly enhanced, and instability of nozzle geometric configuration is enhanced.
It can be obtained from above that maximum deformation of serpentine nozzle within five kinds of length-to-diameter ratios all occurs in downstream area of first bend, wall bulge characteristic is more obvious, bending degree of equal straight section gradually increases.Compared with equal straight section of nozzle, downstream channel of first bend is more susceptible to structural deformation caused by aerodynamic load.And as length-to-diameter ratio increases, deformation distribution range on wall surface of serpentine nozzle gradually expands, but distribution position has no significant difference.

Nozzle aerodynamic performance under different length-to-diameter ratios
The aerodynamic performance results of double serpentine nozzle before and after fluid-structure coupling effect under different length-to-diameter ratios are statistically analyzed.Figure 14 compares aerodynamic performance of serpentine nozzle under different length-to-diameter ratios.In uncoupled state, as length-to-diameter ratio increases, total pressure recovery coefficient and flow coefficient increase continuously, especially when length-to-diameter ratio is 2.2-2.4,there is obvious rising trend, and then shows downward trend.Thrust coefficient increases continuously before length-to-diameter ratio is 2.8, and slightly decreases when length-to-diameter ratio is larger than 2.8.And thrust coefficient increases with increase of length-to-diameter ratio, but change degree is small.In coupled state, total pressure recovery coefficient flow coefficient and thrust coefficient all decrease significantly compared with uncoupled state.When length-to-diameter ratio is 2.4, influence brought by fluid-structure coupling effect is lowest.Total pressure recovery coefficient decreases by 0.40%, flow coefficient decreases by 5.34%, and thrust coefficient decreases by 0.92%.When length-to-diameter ratio is 3, influence brought by fluid-structure coupling effect is highest.At this time, total pressure recovery coefficient decreases by 0.48%, flow coefficient decreases by 5.95%, and thrust coefficient decreases by 1.12%.To analyze the reasons for the changes in the aerodynamic performance of the serpentine nozzle with different length-to-diameter ratios before and after the coupling effect, the flow characteristics inside the serpentine nozzle are first analyzed.Figure 15 shows the static pressure distribution on the symmetrical wall of the serpentine nozzle with different length-to-diameter ratios, where P/P b is the dimensionless ratio of the wall static pressure to the ambient pressure, and x/L is the dimensionless ratio of the nozzle abscissa to the nozzle length.In the uncoupled state, when the length-to-diameter ratio is 2.2, due to the low static pressure on the upper wall of the first bend of the serpentine nozzle, a large adverse pressure gradient is generated downstream, resulting in severe flow separation.The influence of flow separation propagates upstream, reducing the local maximum Mach number at the first bend of the nozzle, increasing the static pressure, and accelerating the airflow to supersonic speed at the entrance of the straight section after passing through the second bend.When the length-to-diameter ratio is greater than 2.2, as the length-to-diameter ratio increases, the static pressure on both the upper wall of the first bend and the lower wall of the second bend of the serpentine nozzle gradually increases.In the coupled state, due to the bulging of both upper and lower walls in middle area of channel downstream of first bend, a local expansion-contraction feature appears in nozzle channel.When length-to-diameter ratio is 2.2, due to nozzle configuration causing excessive curvature at first bend and local expansion deformation of downstream wall at first bend, downstream gas accelerates continuously, Mach number at first bend increases, wall static pressure decreases, and flow separation improves.When length-to-diameter ratio is greater than 2.2, there is no obvious flow separation characteristic in airflow.Due to local expansion deformation of downstream wall at first bend, airflow accelerates gradually in this area.Compared with before coupling, Mach number at first bend decreases slightly.To analyze more specifically the influence of fluid-structure coupling on the aerodynamic performance of serpentine nozzles with different length-to-diameter ratios and the reasons for the performance changes, Figure 16 shows the dimensionless axial positions of the sections along the serpentine nozzle, where: section A is the exit section of the mixing chamber or the inlet section of the nozzle; section B is between the nozzle inlet and the first bend; section C is at the first bend; section D is between the first and second bends; section E is at the second bend; section F is the inlet section of the straight section; section G is the exit section of the nozzle.Take the bending direction of the serpentine nozzle profile as the Y-axis direction, and upward as the positive direction.Figure 17 shows the Mach number distribution on the sections along the serpentine nozzle with different length-to-diameter ratios.According to the above analysis, in the uncoupled state, as the length-to-diameter ratio increases, the longitudinal curvature near the first bend of the nozzle decreases, the degree of flow contraction in the tube decreases, and the degree of flow acceleration weakens.Therefore, the Mach number at sections C and D decreases as the length-to-diameter ratio increases.As the length-to-diameter ratio increases, the longitudinal curvature of the nozzle profile gradually decreases, and the degree of flow acceleration in the local high-speed area near sections C and D weakens, and the supersonic area of the airflow at sections E and F gradually disappears.In the coupled state, when the length-to-diameter ratio is 2.2, due to wall expansion causing intense mixing of airflow on both sides of upper wall at section D compared with uncoupled state low-speed area in middle area of section D decreases and large low-speed areas form on both sides along flow direction transfer to sections E F and G. Before and after bidirectional coupling effect due to local expansion-contraction structure appearing on downstream wall at first bend speed of airflow after first bend increases compared with uncoupled state reaching local supersonic speed in nozzle channel showing Mach number increase from section D to section G reaching supersonic speed at section F Mach number at sections D and E decreases as length-to-diameter ratio increases local acceleration loss at lower wall outlet of second bend decreases change in aerodynamic performance caused by local acceleration loss gradually decreases.Figure 18 shows x-direction vorticity distribution on sections along serpentine nozzles with different length-to-diameter ratios.In uncoupled state difference in length-to-diameter ratio causes downstream flow field disturbance to have little effect on upstream flow characteristics so overall vorticity distribution trend is basically consistent for different lengths-to-diameters When length-to-diameter ratio is 2.2 flow separation occurs on upper wall of first bend due to excessive curvature of profile therefore vorticity at sections downstream of first bend increases significantly When length-to-diameter ratio is greater than 2.2 as length-to-diameter ratio increases longitudinal curvature along nozzle decreases and no severe flow separation occurs therefore overall vorticity distribution trend is basically consistent In coupled state when length-to-diameter ratio is 2.2 due to local expansion deformation of downstream wall at first bend vorticity in middle area of section D decreases while local expansion-contraction feature of wall causes flow velocity on both sides of section D vorticity on both sides of section D increases Due to flow separation effect upper wall vortex at middle of section E intensifies vortex mixing vorticity increases significantly as airflow flows backward upper wall vortex gradually moves to both sides forming two pairs of corner vortices at sections F and G For cases where length-to-diameter ratio is 2.4 to 3 aerodynamic load caused by structural deformation does not cause flow separation downstream of first bend vorticity does not change significantly compared with before bidirectional fluid-structure coupling effect.Figure 19 shows the thrust vector angle at the exit of the serpentine nozzle with different length-to-diameter ratios.In the uncoupled state, the thrust vector angle at the nozzle exit is close to 0°, and the tail jet is sprayed horizontally.In the coupled state, the structure of the nozzle exit section bends upward along the longitudinal direction under the action of aerodynamic load, and the tail jet deflects upward along the axial direction.When the length-to-diameter ratio is 2.2-2.4,the thrust vector angle at the nozzle exit decreases as the length-to-diameter ratio increases; when the length-to-diameter ratio is greater than 2.4, the thrust vector angle at the nozzle exit increases as the length-to-diameter ratio increases.Considering the changes of total pressure recovery coefficient and thrust vector angle, the thrust coefficient decreases, and the change rate gradually increases.According to the above analysis, under fluid-structure coupling effect, friction loss and local acceleration loss of airflow in channel downstream of first bend increase.When length-to-diameter ratio is 2.2, due to flow separation causing large mixing loss, total pressure recovery coefficient and flow coefficient decrease significantly.When length-to-diameter ratio is greater than 2.2, mixing loss is small, comprehensively total pressure recovery coefficient and flow coefficient decrease, and decrease amplitude is smaller than case where length-to-diameter ratio is 2.2.As length-to-diameter ratio increases, local acceleration loss decreases, wall friction loss increases, these flow losses have opposite influence trends and similar influence effects, eventually resulting in similar change amplitude of total pressure recovery coefficient and flow coefficient.Change of thrust coefficient is related to both total pressure recovery coefficient and thrust vector angle.As length-to-diameter ratio increases, thrust vector angle of nozzle first decreases then increases.Considering influence of total pressure recovery coefficient and thrust vector angle, when length-to-diameter ratio is less than 2.6, change rate of thrust coefficient increases; when length-to-diameter ratio is greater than 2.6, change rate of thrust coefficient is similar.

Conclusion
In this paper, based on serial bidirectional loose coupling method, influence of different length-to-diameter ratios on bidirectional fluid-structure coupling characteristics of turbofan engine double serpentine nozzle is studied with double Serpentine nozzle of turbofan engine as research object.
Main conclusions are as follows: 1) Structural deformation characteristics of serpentine nozzle are mainly located on upstream channel and upper wall of outlet equal section of first bend.As length-to-diameter ratio increases, due to influence of first bend channel shape outlet equal section shape gradually becoming "flattened" and serpentine nozzle bending degree gradually increasing deformation amount of upper and lower wall areas downstream of first bend gradually increases while deformation amount of upper wall at outlet first increases then decreases Maximum deformation of upper and lower walls of serpentine nozzle occurs on downstream wall at first bend.
2) Fluid-structure coupling effect has great influence on nozzle aerodynamic performance total pressure recovery coefficient flow coefficient and thrust coefficient all decrease significantly When length-to-diameter ratio is 2.2 due to large flow loss caused by flow separation total pressure recovery coefficient decreases by 0.77% flow coefficient decreases by 5.80% thrust coefficient decreases by 1.13% When length-to-diameter ratio is greater than 2.4 as length-to-diameter ratio increases aerodynamic performance change caused by fluid-structure coupling effect gradually increases reaching maximum when length-to-diameter ratio is 3 total pressure recovery coefficient decreases by 0.48% flow coefficient decreases by 5.95% thrust coefficient decreases by 1.12%.
3) When length-to-diameter ratio is 2.2 and 3 fluid-structure coupling effect has great influence on local deformation and aerodynamic performance of serpentine nozzle In case of small length-to-diameter ratio due to large mixing loss caused by flow separation aerodynamic performance decreases when length-to-diameter ratio is between 2.2-2.4 nozzle bends upward along axial direction angle decreases thrust vector angle decreases In case of large length-to-diameter ratio due to nozzle configuration causing excessive local deformation large mixing loss and local acceleration loss are generated which leads to decrease in aerodynamic performance Nozzle bends upward along axial direction angle gradually increases as length-to-diameter ratio increases thrust vector angle gradually increases.

Figure 1 .
Figure 1.Geometry model of double serpentine nozzle.

Figure 2 .
Figure 2. Design parameters of double serpentine nozzle.

Figure 3 .
Figure 3. Criterions to completely shield high temperature.

Figure 5 .
Figure 5. Numerical grid and boundary condition.Figure 6. Finite element model of the solid domain.

Figure 6 .
Figure 5. Numerical grid and boundary condition.Figure 6. Finite element model of the solid domain.

Figure 7 .
Figure 7. Solution mechanism of series loosely two-way coupled algorithm.

Figure 8 .Figure 9 .
Figure 8. Experimental model of serpentine nozzle and wall spraying to measure scattering effects.Numerical simulation results and experimental measurement data are compared and analyzed for nozzle wall surface deformation displacement distribution and wall surface profile on symmetry plane.Comparison of deformation displacement for upper/lower wall surfaces of serpentine nozzle is shown in Figure 9.It can be seen from figure that key deformation characteristics and distribution positions of upper/lower wall surfaces of nozzle obtained by fluid-structure coupling numerical calculation are basically consistent with experimental results, mainly showing local "bulge" of first bend channel and upward offset of nozzle equal straight section along Y direction.Main difference between numerical simulation results and experimental measurement data is at nozzle outlet position.Y-direction deformation displacement values of upper/lower wall surfaces at nozzle outlet measured by experiment are slightly larger than numerical prediction results.

Figure 10 compares
Figure 10 compares numerical simulation results and experimental measurement data of inner wall surface profile on symmetry plane of serpentine nozzle model, Table5shows the contour error of symmetry surface of serpentine nozzle model.Areas with large error between numerical prediction value and experimental measurement value are mainly located on upper wall surface of downstream channel of first bend and nozzle outlet position.Maximum relative error value at position of upper wall surface of downstream channel of first bend is 9.0%, maximum relative error value of upper wall surface at nozzle outlet is 8.8%, and maximum relative error value of lower wall surface at nozzle outlet is 9.1%.Compared with numerical/experimental errors of different types of nozzles at home and abroad[25,26] , where average error in literature[25] is 20%, minimum error in literature[26] is 7%, and error value

Figure 10 .
Figure 10.Comparison of wall profiles on the symmetry plane be-tween numerical simulation and experiment.

Figure 12 .
Figure 12.Displacement contours of the upper wall of serpentine nozzle under different length-to-diameter ratios.

Figure 13 .
Figure 13.Displacement contours of the lower wall of serpentine nozzle under different length-to-diameter ratios.

Figure 14 .
Figure 14.Comparisons of aerodynamic performance of serpentine nozzle under different length-to-diameter ratios.

Figure 15 .
Figure 15.Comparisons of much number contours inside serpentine nozzle under different length-to-diameter ratios for two cases.

Figure 16 .
Figure 16.Streamwise locations of the cross sections inside serpentine nozzle.

Figure 17 .
Figure 17.Comparisons of the Mach number on the cross sections inside serpentine nozzle under different length-to-diameter ratios for two cases.

Figure 18 .
Figure 18.Comparisons of the x vorticities on the cross sections inside serpentine nozzle under different length-to-diameter ratios for two cases.

Figure 19 .
Figure 19.Thrust vector angle of the serpentine nozzle exit under different length-to-diameter ratios for two cases.

Table 1 .
Design parameters of serpentine nozzle.

Table 3 .
The physical properties of solid material.
Δt on coupling surface from structural response characteristics obtained by ABAQUS finite element analysis, and transfers it to corresponding flow field boundary grid, causing change of flow field boundary coordinates; ⑤Fluent completes unsteady calculation according to new flow field boundary, and obtains flow field at moment; ⑥Repeat steps ②-⑤ within Δt~2Δt time step, and take flow field distribution obtained by coupling calculation of 0~Δt time step as initial field value of this time step.
MpCCI (Mesh-based parallel Code Coupling Interface) is used as the data coupling exchange platform for fluid domain and solid domain, realizing the mutual transfer of aerodynamic load F obtained by calculating fluid domain and deformation displacement U obtained by calculating solid domain on fluid-structure coupling surface.Among them, aerodynamic load F includes pressure and viscous force generated by flow.The time domain coupling advancement process within 0~Δt time step is as follows: ①Before coupling solution, that is, at t=0 moment, obtain steady-state flow field solution under given boundary conditions; ②MpCCI extracts aerodynamic load FΔt on coupling surface from steady-state flow field obtained by Fluent calculation, and applies it to corresponding structure surface element; ③Abaqus completes transient structural analysis according to external aerodynamic load, and obtains deformation U Δt distribution at moment Δt; ④ MpCCI extracts deformation U

Table 4 .
Coupling time step verification results.

Table 5 .
Contour error value of symmetric surface.

Table 6 .
Values of longitudinal offset distance under different length-to-diameter ratios.length-to-diameter ratios-L/D Longitudinal deflection of the second bend-ΔY 2 /L 2