The e-N method is widely used in transition prediction. The amplitude growth rate used in the e-N method is usually provided by the linear stability theory （LST） based on the local parallel hypothesis. Considering t...The e-N method is widely used in transition prediction. The amplitude growth rate used in the e-N method is usually provided by the linear stability theory （LST） based on the local parallel hypothesis. Considering the non-parallelism effect, the parabolized stability equation （PSE） method lacks local characteristic of stability analysis. In this paper, a local stability analysis method considering non-parallelism is proposed, termed as EPSE since it may be considered as an expansion of the PSE method. The EPSE considers variation of the shape function in the streamwise direction. Its local characteristic is convenient for stability analysis. This paper uses the EPSE in a strong non-parallel flow and mode exchange problem. The results agree well with the PSE and the direct numerical simulation （DNS）. In addition, it is found that the growth rate is related to the normalized method in the non-parallel flow. Different results can be obtained using different normalized methods. Therefore, the normalized method must be consistent.展开更多
An improved expansion of the parabolized stability equation(iEPSE) method is proposed for the accurate linear instability prediction in boundary layers. It is a local eigenvalue problem, and the streamwise wavenumber ...An improved expansion of the parabolized stability equation(iEPSE) method is proposed for the accurate linear instability prediction in boundary layers. It is a local eigenvalue problem, and the streamwise wavenumber α and its streamwise gradient dα/dx are unknown variables. This eigenvalue problem is solved for the eigenvalue dα/dx with an initial α, and the correction of α is performed with the conservation relation used in the PSE. The i EPSE is validated in several compressible and incompressible boundary layers. The computational results show that the prediction accuracy of the i EPSE is significantly higher than that of the ESPE, and it is in excellent agreement with the PSE which is regarded as the baseline for comparison. In addition, the unphysical multiple eigenmode problem in the EPSE is solved by using the i EPSE. As a local non-parallel stability analysis tool, the i EPSE has great potential application in the eNtransition prediction in general three-dimensional boundary layers.展开更多
The parabolized stability equation （PSE） method has been proven to be a useful and convenient tool for the investigation of the stability and transition problems of boundary layers. However, in its original formulat...The parabolized stability equation （PSE） method has been proven to be a useful and convenient tool for the investigation of the stability and transition problems of boundary layers. However, in its original formulation, for nonlinear problems, the complex wave number of each Fourier mode is determined by the so-called phase-locked rule, which results in non-self-consistency in the wave numbers. In this paper, a modification is proposed to make it self-consistent. The main idea is that, instead of allowing wave numbers to be complex, all wave numbers are kept real, and the growth or decay of each mode is simply manifested in the growth or decay of the modulus of its shape function. The validity of the new formulation is illustrated by comparing the results with those from the corresponding direct numerical simulation （DNS） as applied to a problem of compressible boundary layer with Mach number 6.展开更多
This study is to numerically test the interfacial instability of ferrofluid flow under the presence of a vacuum magnetic field.The ferrofluid parabolized stability equations(PSEs)are derived from the ferrofluid stabil...This study is to numerically test the interfacial instability of ferrofluid flow under the presence of a vacuum magnetic field.The ferrofluid parabolized stability equations(PSEs)are derived from the ferrofluid stability equations and the Rosensweig equations,and the characteristic values of the ferrofluid PSEs are given to describe the ellipticity of ferrofluid flow.Three numerical models representing specific cases considering with/without a vacuum magnetic field or viscosity are created to mathematically examine the interfacial instability by the computation of characteristic values.Numerical investigation shows strong dependence of the basic characteristic of ferrofluid Rayleigh-Taylor instability(RTI)on viscosity of ferrofluid and independence of the vacuum magnetic field.For the shock wave striking helium bubble,the magnetic field is not able to trigger the symmetry breaking of bubble but change the speed of the bubble movement.In the process of droplet formation from a submerged orifice,the collision between the droplet and the liquid surface causes symmetry breaking.Both the viscosity and the magnetic field exacerbate symmetry breaking.The computational results agree with the published experimental results.展开更多
基金Project supported by the National Natural Science Foundation of China(Nos.11332007,11172203,and 91216111)
文摘The e-N method is widely used in transition prediction. The amplitude growth rate used in the e-N method is usually provided by the linear stability theory （LST） based on the local parallel hypothesis. Considering the non-parallelism effect, the parabolized stability equation （PSE） method lacks local characteristic of stability analysis. In this paper, a local stability analysis method considering non-parallelism is proposed, termed as EPSE since it may be considered as an expansion of the PSE method. The EPSE considers variation of the shape function in the streamwise direction. Its local characteristic is convenient for stability analysis. This paper uses the EPSE in a strong non-parallel flow and mode exchange problem. The results agree well with the PSE and the direct numerical simulation （DNS）. In addition, it is found that the growth rate is related to the normalized method in the non-parallel flow. Different results can be obtained using different normalized methods. Therefore, the normalized method must be consistent.
基金Project supported by the National Natural Science Foundation of China(Nos.11332007,11402167,11672205,and 11732011)the National Key Research and Development Program of China(No.2016YFA0401200)
文摘An improved expansion of the parabolized stability equation(iEPSE) method is proposed for the accurate linear instability prediction in boundary layers. It is a local eigenvalue problem, and the streamwise wavenumber α and its streamwise gradient dα/dx are unknown variables. This eigenvalue problem is solved for the eigenvalue dα/dx with an initial α, and the correction of α is performed with the conservation relation used in the PSE. The i EPSE is validated in several compressible and incompressible boundary layers. The computational results show that the prediction accuracy of the i EPSE is significantly higher than that of the ESPE, and it is in excellent agreement with the PSE which is regarded as the baseline for comparison. In addition, the unphysical multiple eigenmode problem in the EPSE is solved by using the i EPSE. As a local non-parallel stability analysis tool, the i EPSE has great potential application in the eNtransition prediction in general three-dimensional boundary layers.
基金supported by the National Natural Science Foundation of China(Nos.11202147,11472188,11332007,11172203,and 91216111)the Specialized Research Fund(New Teacher Class)for the Doctoral Program of Higher Education(No.20120032120007)
文摘The parabolized stability equation （PSE） method has been proven to be a useful and convenient tool for the investigation of the stability and transition problems of boundary layers. However, in its original formulation, for nonlinear problems, the complex wave number of each Fourier mode is determined by the so-called phase-locked rule, which results in non-self-consistency in the wave numbers. In this paper, a modification is proposed to make it self-consistent. The main idea is that, instead of allowing wave numbers to be complex, all wave numbers are kept real, and the growth or decay of each mode is simply manifested in the growth or decay of the modulus of its shape function. The validity of the new formulation is illustrated by comparing the results with those from the corresponding direct numerical simulation （DNS） as applied to a problem of compressible boundary layer with Mach number 6.
基金the National Natural Science Foundation of China(No.11971411)the Research Foundation of Education Bureau of Hunan Province of China(No.18A067)。
文摘This study is to numerically test the interfacial instability of ferrofluid flow under the presence of a vacuum magnetic field.The ferrofluid parabolized stability equations(PSEs)are derived from the ferrofluid stability equations and the Rosensweig equations,and the characteristic values of the ferrofluid PSEs are given to describe the ellipticity of ferrofluid flow.Three numerical models representing specific cases considering with/without a vacuum magnetic field or viscosity are created to mathematically examine the interfacial instability by the computation of characteristic values.Numerical investigation shows strong dependence of the basic characteristic of ferrofluid Rayleigh-Taylor instability(RTI)on viscosity of ferrofluid and independence of the vacuum magnetic field.For the shock wave striking helium bubble,the magnetic field is not able to trigger the symmetry breaking of bubble but change the speed of the bubble movement.In the process of droplet formation from a submerged orifice,the collision between the droplet and the liquid surface causes symmetry breaking.Both the viscosity and the magnetic field exacerbate symmetry breaking.The computational results agree with the published experimental results.
