On a general theory for compressing process and aeroacoustics: linear analysis

Transcription

1 Acta Mech Sin (2010) 26: DOI /s RESEARCH PAPER On a general theory for compressing process and aeroacoustics: linear analysis F. Mao Y. P. Shi J. Z. Wu Received: 4 September 2009 / Revised: 26 October 2009 / Accepted: 4 December 2009 / Published online: 13 February 2010 The Chinese Society of Theoretical and Applied Mechanics and Springer-Verlag GmbH 2010 Abstract Of the three mutually coupled fundamental processes (shearing, compressing, and thermal) in a general fluid motion, only the general formulation for the compressing process and a subprocess of it, the subject of aeroacoustics, as well as their physical coupling with shearing and thermal processes, have so far not reached a consensus. This situation has caused difficulties for various in-depth complex multiprocess flow diagnosis, optimal configuration design, and flow/noise control. As the first step toward the desired formulation in fully nonlinear regime, this paper employs the operator factorization method to revisit the analytic linear theories of the fundamental processes and their decomposition, especially the further splitting of compressing process into acoustic and entropy modes, developed in 1940s 1980s. The flow treated here is small disturbances of a compressible, viscous, and heat-conducting polytropic gas in an unbounded domain with arbitrary source of mass, external body force, and heat addition. Previous results are thereby revised and extended to a complete and unified theory. The theory provides a necessary basis and valuable guidance for developing corresponding nonlinear theory by clarifying certain basic issues, such as the proper choice of characteristic variables of compressing process and the feature of their governing equations. The project was supported by the National Basic Research Program of China (2009CB724100). F. Mao Y. P. Shi J. Z. Wu (B) State Key Laboratory for Turbulence and Complex Systems, College of Engineering, Peking University, Beijing, China jzwu@coe.pku.edu.cn J. Z. Wu University of Tennessee Space Institute, Tullahoma, TN 37388, USA Keywords Longitudinal (compressing) process Transversal (shearing) process Acoustic mode Entropy mode Aeroacoustics 1 Introduction It has been commonly believed that a general viscous and heat-conducting compressible flow contains three fundamental processes: the longitudinal or compressing process, the transverse or shearing process, and the thermodynamic or entropy process. More precisely, the application of the Helmholtz decomposition (HD) to the momentum equation first defines and distinguishes the compressing and shearing processes [1]; then once being combined with the continuity and energy equations, the former is enlarged to include the entropy process. These three fundamental processes are governed by equations of different types with different dimensionless parameters (e.g., Mach number and Reynolds number), and hence have very different physical evolution behaviors; but they are coupled both inside the flow mainly via the nonlinearity of the equations, and on fluid boundary via the velocity adherence and heat-transfer conditions. A full understanding of these processes and their coupling mechanisms is the very basis for complex multi-process flow diagnosis and optimal configuration design, as well as for noise alleviation and control in various engineering applications. While the shearing process and its interaction with the other two processes have been well formulated in the realm of vorticity and vortex dynamics [1], and the formulation of thermal process is also clear, the general formulation for the compressing process and its important subprocess, the

2 356 F. Mao et al. aeroacoustics, still remains in controversy. 1 This situation has made it difficult to identify the physical sources and understand the nonlinear evolution of the compressing process in a generic viscous and heat-conducting shear (vortical) flow. The root of such a situation may be traced in part to the rich but multiple-road development of aeroacoustics. The basic framework of this branch was settled by Rayleigh [2] for whom the HD was the main tool to identify the sound wave (For a detailed review of Rayleigh s theory in the context of aeroacoustics see Doak [3]), which has been followed by Lagerstrom et al. [4], Wu [5], Chu and Kovasznay [6], and Pierce [7], among others, in their analytical studies on the linearized or iterative theory of fundamental processes and their splitting from late 1940s 1980s. But this line of development was overwhelmed by Lighthill s ingenious acoustic analogy theory [8] which puts the HD aside and represents the local mechanisms responsible for sound generation by a set of equivalent quadrupole sources in a fictitious and otherwise undisturbed medium where the sound speed is constant. Owing to the ability of analytically solving the acoustic part of the problem, this theory has since been rapidly developed into a powerful means to predict the far-field acoustic behavior of various unsteady Navier Stokes flows with stationary and moving solid boundaries, and applied successfully to many engineering problems. Today the theory still remains to be a major basis of modern computational aeroacoustics [9]. However, it is the very nature of the analogy that makes Lighthill s theory inherently impossible to clearly distinguish the physical mechanisms influencing sound propagation (such as refraction and diffraction) in an actual flow and those serving as physical sources of sound, as well as the reaction of the acoustic field to the background flows (for this kind of comments see, e.g., Doak [10] and Yates [11]). This situation evidently demands to be improved in view of the aforementioned need for complex flow diagnosis, configuration design, and noise alleviation. The above unsatisfactory situation is also closely tied with the very nature of the longitudinal process itself, well beyond aeroacoustics. In contrast to the shearing process characterized unambiguously by the vorticity and governed by a single vectorial convective-diffusive equation, the compressing process involves not only the momentum equation but also the continuity and energy equations. Consequently, quite a few fluid dynamic and thermodynamic scalar variables could serve alternatively as the candidate for characterizing the process, with their governing equations varying in form and order, but so far no single representative equation (as the counterpart of the vorticity transport equation) has been commonly accepted. 1 Other subclasses of compressing process include, for example, the shock formation and viscous shock-layer theory, and the classic inviscid and irrotational high-speed aerodynamics. We believe that what has currently been lacking is a general and complete theoretical formulation for compressing process, including aeroacoustics, as the counterpart of vorticity and vortex dynamics for shearing process. The formulation should lead to theories able to cover things beyond the reach of existing ones (these include, in the context of aeroacoustics, nonlinear acoustic theory without any shear flow and Lighthill s acoustic analogy). A natural road along this line is evidently returning to the HD-based track. As the first step of this returning, in the present paper we revisit and further clarify the structure of aforementioned linearized theories for the fundamental processes in a compressible, viscous, and heat-conducting fluid motion. Our focus is on the compressing process and its subprocesses, the so-called acoustic mode and entropy mode [6]. The importance of the linearized theory for further developing the fully nonlinear theories lies in the fact that the theory of the fundamental processes and their splitting can be analytically formulated, and hence provides a complete and unambiguous prototype as the basis of constructing nonlinear theory, which has to be fully consistent with the linear one and includes the latter naturally as the lowest-order approximation. Thus, firstly, the variables qualified to characterize compressing process and its submodes in any nonlinear theory have to be first qualified in the linear theory; and secondly, the number of independent variables and the order of possible governing equation(s) for compressing process in linear theory may provide important clue in constructing nonlinear theory. Here, for using the linear theory as the starting point of developing nonlinear theories, rather than being confined to purely linearized processes themselves, it is of crucial importance to permit arbitrary source of mass, external body force, and heat addition in the governing equations, because these inhomogeneous terms can stand for a symbolic representation of the corresponding nonlinear terms. The technical approach throughout this paper is operator factorization. Although analysis in spectral space can be simplified to algebraic equations and lead to various dispersion relations [7], the mathematical nature of the governing differential equations is no as clear as the operator analysis in physical space. If the high-order differential operator governing a multiprocess flow can be factorized, then the processes or modes in the flow are physically decoupled, and otherwise are coupled. A full decoupling of the processes or modes requires that the factorization has to be applied not only to the operator governing relevant variables but also to the source terms simultaneously. As will be seen, in so doing some previous conclusions have to be revised or put in more appropriate places, and new results will be obtained. Consequently, a complete picture is achieved that unifies all previous results into a whole. In what follows we first review in Sect. 2 the basic equations of the fundamental processes and the previous results on

3 On a general theory for compressing process and aeroacoustics 357 process splitting, in both nonlinear and linearized theories. We then derive the linear differential operator of a single high-order compressing equation for any compressing variables with arbitrary inhomogeneous terms, and examine the general condition for the existence of acoustic wave (Sect. 3). Our operator analysis is continued in Sect. 4, where the conditions for mode splitting are further identified and equations for each mode are analyzed. Section 5 makes a few conclusions and discussions. A fully general and nonlinear formulation for compressing process and aeroacoustics will be reported elsewhere. 2 Background and previous works 2.1 Fundamental equations and process splitting We start from the basic equations of polytropic gas with arbitrary mass addition m, external body force f, and heat addition Q per unit mass. Let μ and λ be the first (shear) and second viscosities, respectively, which are assumed constant in this work. Denote the dilatation and vorticity by ϑ = u and ω = u, respectively. Then from the viscous stress tensor V λϑi + 2μD : D, with I being the unit tensor and D the strain-rate tensor, one may derive [1] 1 ρ V = ν θ ϑ ν ω, where ν = μ ρ, ν θ = 1 (λ + 2μ) (1) ρ are the shear and longitudinal kinematic viscosities, respectively. With this little preparation, the desired set of fundamental equations reads, by writing T s = T 0 s + (T T 0 ) s where T 0 = 0 is a reference temperature and only the second term is nonlinear, t ρ + u ρ + ρϑ = ρm, t u +L + (H T 0 s) ν θ ϑ +ν ω = f, ρt ( t s + u s) + q = ρ Q, p = ρ RT, where R = c p c v is the gas constant, and (2a) (2b) (2c) (2d) H h q2 with q = u, (2e) L ω u (T T 0 ) s, D : V = λϑ 2 + 2μD : D (2f) (2g) are the total enthalpy, generalized Lamb vector and dissipation, respectively. The m, f, Q terms can be either actual sources or account for nonlinear interactions, such as that in the successive approximation scheme proposed by Chu and Kovasznay [6]. As will be shown later, these terms are important to identify different modes in compressing process. If they are set zero, then the result is only meaningful in the purely linear flow, which would offer no help to the study of realistic nonlinear flow. We now take the curl and divergence of Eq. (2b). This yields the respective governing equations for the shearing and compressing processes: ( t ω ν 2 )ω = (L f ) ν ( ω )+ ν θ ϑ, (3a) ( t ν θ 2 )ϑ + 2 (H T 0 s) = L + ν θ ϑ ν ( ω ), (3b) where the right-hand side as well as q 2 /2 on the left-hand side is nonlinear terms. Equation (3a) is the basic equation for vorticity dynamics. Equation (3b), along with Eq. (2a), (2b), and (2d), describes the compressing process, which is the major concern of this paper. Although scalar equations are simpler than a vector equation like Eq. (3a) in some aspects, the fact that multiple scalar variables appearing in these equations, e.g., ϑ, h, q 2 /2, s, has suggested that the compressing process has very distinct feature and yet many issues are to be studied. This is why we engaged ourselves to the simplest linearized flow model first, as developed in this paper. 2.2 Process splitting in unbounded linearized flows Following Wu [5], we consider small disturbances to an unbounded fluid otherwise at rest with constant (u, p, s) = (0, p 0, s 0 ), where the pressure p and entropy s are chosen as independent thermodynamic variables. Wu [5] uses temperature T rather than entropy s as the thermodynamic variable, but the results are not affected. Introduce a small parameter ɛ 1 and define the disturbance quantities by u = U 0 (ɛu + ), p = γ p 0 (1/γ + ɛp + ), h = c p T 0 (1 + ɛh + ), ρ = ρ 0 (1 + ɛρ + ), s = c p (ɛs + ), (4) ν = ν 0 (1 + ɛν + ), ν θ = ν θ0 (1 + ɛν + ), k = k 0 (1 + ɛk + ), Q = h 0 (ɛ Q + ),

4 358 F. Mao et al. where the primed variables are dimensionless, there is ρ = p s, T = h = γ p ρ = (γ 1)p + s. (5) To reduce the number of unknowns, recall that for polytropic gas the continuity equation and equation of state can be expressed as ϑ = 1 Ds c p Dt 1 Dp ρc 2 Dt dt T = ds + γ 1 c p γ dp p, = 1 R Ds Dt 1 c 2 Dh Dt, (6a) (6b) where c is the speed of sound. Substitute Eq. (4) into Eq. (2a), (2b) and use Eq. (6), to the leading O(ɛ) terms it follows that t (p s ) + ϑ = m, t u + c 2 p ν θ0 ϑ + ν 0 ω = f, ( t κγ ) 2 s β 2 p = Q, (7a) (7b) (7c) where c 2 p = h 0 (h s ), Q is scaled by ɛc p T 0, κ = k 0 = γν (8a) c v ρ 0 Pr is the thermometric conductivity, and β (γ 1) κ γ. (8b) In what follows we save the prime for neatness and introduce the Helmholtz decomposition for of u and f, denoted here as u = v + φ, u = v = ω, u = 2 φ = ϑ, f = f + f = f + χ, f = f, f = f = 2 χ. (9a) (9b) In the linear regime, the quadratic term L in Eq. (3a) and (3b) is neglected so that the shearing process and compressing process are decoupled. In fact, as observed by Wu [5], the curl of Eq. (7b) splits at once the shearing process from the rest of the fluid motion, governed by the linearized diffusion equation for the vorticity or solenoidal velocity: ( t ν 2 )ω = f, or (10a) ( t ν 2 )v = f, (10b) which is completely independent of Eq. (7a) and (7c). Subtract Eq. (10b) from Eq. (7b), the linearized governing equations of the compressing process follow (e.g., Ref. [5]): t (p s) + ϑ = m, (11a) ( t ν θ 2 )φ + c 2 p = χ, (11b) ( t κγ ) 2 s β 2 p = Q. (11c) This leads to the classical splitting theorem for linearized compressible and viscous flow, given first by Lagerstrom et al. [4] for non-conducting fluid and then generalized by Wu [5] to include heat conduction: Spliting Theorem For small-disturbed flow of a polytropic gas in an unbounded domain at rest at infinity and with arbitrarily given source of mass m, external body force f = f + χ, and heat addition Q, a flow (u, p, s) governed by the linear system (7) can be expressed as the sum of a shearing process (v, 0, 0) and a compressing process ( φ, p, s), such that (v, 0, 0) with transverse force f and m = Q = 0 satisfy Eq. (10), ( φ, p, s) with longitudinal force χ and m, Q satisfy Eq. (11), and u = v + φ. The decomposition is unique provided the initial values of v and φ are separately given. Note that in the initial condition one usually prescribes the full velocity u, but it can be uniquely decomposed to v and φ because the boundary condition at infinite excludes any harmonic function that could otherwise make the decomposition non-unique. 2.3 Modes splitting of the compressing process The splitting theorem enables us to focus on the compressing process alone, which is our present concern. If all of m,χ,q vanish, Pierce [7] has shown that, when the viscosities and conductivity are small parameters, the compressing process can be further decomposed into an acoustic mode, which is characterized by pressure and effectively isentropic, and an entropy mode, which is characterized by entropy and effectively isobaric. We shall adopt this concept for our mode splitting. However, we insist on retaining all the source terms as our ultimate concern is developing a nonlinear theory. For a special case when χ, Q are non-zero, Wu [5] has noticed that when Pr θ = γν θ0 /κ = 1, the pressure can enjoy being governed by a reduced-order heat-conducting wave equation. The same assumption on Pr θ is followed by Chu and Kovasznay [6], where they added the mass-production term m. The complete form of this pressure equation reads { 2 t (c2 +κ t ) 2 }p =( t ν θ 2 )m 2 χ + t Q. (12) Due to this nice property, Chu and Kovasznay [6] decomposed the compressing process into a sound mode and an

5 On a general theory for compressing process and aeroacoustics 359 entropy mode. Note that the sound mode in their work is not isentropic as we require for it being an acoustic mode. Indeed, it is easy to show that an independent entropy mode can exist only if both χ and Q vanish, otherwise it is a by-product of the acoustic mode. Since the condition Pr θ = 1 is too restrictive, theories under this condition are not fully realistic. Thus, this assumption will be removed in the following development. with t 2 t L M = c 2 D θ 0, β 2 0 D κ/γ p m V = φ, S = χ, s Q (14b) 3 Linearized compressing equations We now proceed to further developing the linear theory for the compressing process governed by system (11). Although the algebra could be greatly simplified if we transformed the linear systems from physical space to the spectral space, as done by Pierce [7] in his book, we perform our calculations in the physical space because the results obtain thereby will be more instinctive for our purpose. We shall employ the method of operator factorization systematically. In so doing various linear differential operators will emerge, so for neatness we introduce some notations. A high-order linear operator applicable to the single equation for all variables will be denoted by L. Besides, we shall meet a few diffusion operators D α and hyperbolic viscous/heatconductive wave operators W η, 2 with α and η denoting specific viscosity/heat-conductivity coefficients: D θ = t ν θ 2, D κ = t κ 2, D κ/γ = t κ γ 2, D β = t β 2 ; W θ = 2 t (c 2 + ν θ t ) 2, W γθ = 2 t (c 2 + γν θ t ) 2, W κ = 2 t (c 2 + κ t ) 2, W b = 2 t (c 2 + b t ) 2, where in W b, b β + ν θ = κ κ γ + ν θ. 3.1 The (p,φ,s) formulation (13a) (13b) (13c) Set u = φ in Eq. (11b), so that the equation can be integrated once and the matrix form of the system is L M V = S, (14a) 2 In the theory of partial differential equations, this kind of wave operators is classified as parabolic. However, due to the smallness of the coefficient of the highest derivative, the sub-characteristics (inviscid characteristics) is more important than the viscous characteristics for most cases [4]. Thus we regard it as a hyperbolic wave operator. in which D νθ and D κ/γ are defined in Eq. (13). System (14) solves three unknowns at proper initial-boundary conditions by three scalar equations. It is well posed. We first seek its decoupled form, i.e., derive the governing equations for each variables. Multiplying this matrix equation from the left by the adjoint of L M,sayL M, casts the system to LV = L M S, where ( ) b L = D κ/γ W b κ γ ν θ t 4 is a high-order linear operator, and D κ D θ 2 D κ/γ t D θ L M = c 2 D κ/γ t D κ c 2 t β D θ 2 β 4 W θ, (15a) (15b) (15c) with D κ/γ and W b having been defined in Eq. (13). From Eq. (15a) it is evident that when the source terms are absent, ( p, φ, s) will satisfy the same homogeneous equation. And so will other variables such as ρ and h since they are simply linear combinations of (p, s). Therefore, the distinctive roles of different variables can only be identified through their special source terms. 3.2 The (h,φ,s) and (h,φ)formulations Alternative to the (p,φ,s) formulation, since c 2 is directly related to enthalpy h, it is also desired to choose (h, s) rather than (p, s) as the independent state variables. Using the second expression for ϑ in Eq. (6a), one can obtain the linearized basic equations in terms of (u, h, s), for which the conclusion of splitting theorem remains effective. Then the linearized compressing equations may alternatively read 1 γ 1 t(h γ s) + 2 φ = m, (16a) ( t ν θ 2 )φ + h 0 (h s) = χ, (16b) t s κ γ 2 h = Q, (16c) of which the matrix form is L M V = S (17a)

6 360 F. Mao et al. with 1/(γ 1) t 2 γ/(γ 1) t L M = h 0 D θ h 0, γ κ 2 0 t h m V = φ, S = χ. s Q (17b) (17c) In fact, one can simply replace p by (h s)/γ in Eq. (14) and the same result follows. The determinant of the matrix is L = detl M = 1 γ 1 { t 3 (c2 +κ t +ν θ t ) t 2 +κ(c 2 /γ +ν θ t ) 4} = 1 γ 1 detl M, as being anticipated. The system (17a) can be further cast to L V = L M S, where (17d) (18a) t D θ t 2 W γθ /(γ 1) L M = c 2 D κ/γ /(γ 1) t D κ /(γ 1) c 2 t /(γ 1), κ/γ D θ 2 κ/γ 4 W θ /(γ 1) (18b) where W γθ is also defined in Eq. (13). It is remarkable that this (h,φ,s) formulation can be further simplified to an (h,φ) formulation or equivalently a (c 2,φ)formulation. To see this we first cast Eq. (17a) to t (γ 1) 2 γ t h (γ 1)m t 1/h 0 D θ t t γ κ 2 0 t φ s = 1/h 0 t χ Q Using the (s, h) relation in the third line, we easily eliminate s from the first and second lines to obtain a system for (h,φ) only: [ Dκ (γ 1) 2 h 0 D κ/γ D θ t ][ ] h = φ [ (γ 1)m + γ Q t χ + h 0 Q ].. (19) This linear system is fully equivalent to those in the (p,φ,s) and (h,φ,s) formulations without any further approximation. Thus we have Lh = D θ t {(γ 1)m + γ Q} (γ 1) 2 { t χ + h 0 Q} = (γ 1)(D θ t m 2 t χ)+ W γθ Q, (20a) Lφ = h 0 D κ/γ {(γ 1)m + γ Q}+ D κ { t χ + h 0 Q} = c 2 ( D κ/γ m + t Q) + D κ t χ, (20b) where L is defined by Eq. (15b). We further notice that the disturbance ɛh = (c 2 c 2 0 )/ (γ 1) merely represents the variation of the squared sound speed, which has been omitted in the present linear theory because c 2 appears always as a coefficient of O(ɛ) variables. Thus, the equation for h is redundant and will not be visited any more. Even in nonlinear theory the equation for h should not be expected to characterize acoustic waves either, but rather be solved jointly with the convective-wave equation in which c 2 appears as an unknown variable. The main result of the preceding analysis can be summarized as follows. Under the same condition as the Splitting Theorem, the following results hold for general linearized compressing process with sound-speed variation neglected: 1. All the flow variables p, h,φ and s satisfy a fifth-order linear differential equation Lg = f g (m,χ,q) with the same homogeneous part but variable-dependent inhomogeneous part. 2. Only a single variable φ suffices to represent the full linearized compressing process, governed by Lφ = f φ (m,χ,q). 3. Generally L cannot be factorized into the product of lower-order operators so that a rigorous splitting of submodes of the compressing process does not exist, each governed by a lower-order equation. 4 Compressing mode decomposition Although we have just shown that the compressing process can simply amount to a single high-order equation Lφ = f φ (m,χ,q), our main concern is the special conditions under which the acoustic mode and entropy mode are separable, each governed by a lower-order equation. Before proceed, we notice that in a viscous fluid any sound wave is always associated with a weak entropy creation and transport. Thus, there has been no rigorous entropy-free viscous acoustic mode. The triple-mode concept (including vortical mode ) was initiated by Chu and Kovasznay [6], but in their theory the three modes were assumed of the same order for a unified iterative procedure, which is however not always the case. Intuitively, a general guiding rule for the distinction of the acoustic and entropy modes could be like this: the entropy variation in the acoustic mode is small, or the acoustic mode is effectively homentropic, and the pressure variation in the

7 On a general theory for compressing process and aeroacoustics 361 entropy mode is small, or the entropy mode is effectively isobaric. But as will be seen below, this rule cannot be always satisfied due to the complexity of the compressing process. Nevertheless, to be able to split the compressing process into acoustic and entropy modes, it is necessary that both the differential operator L and the source terms can be simultaneously factorized. From Eq.(15b) it is evident that L can be factorized if the second term therein vanishes. But the conditions for the possibility of simultaneous factorization of the source terms vary as the chosen variables. This latter fact is important since it implies that different variables are not equally qualified to describe the acoustic mode. 4.1 Inviscid and non-heat-conducting flow For completeness we first remark that when the flow is inviscid and without heat conduction, p and φ simply satisfy the classic wave equation ( 2 t c 2 2 )p = t m 2 χ, (21a) ( 2 t c 2 2 )φ = t χ c 2 m. (21b) The dimensionless energy equation is simply reduced to t s = Q and the entropy is decoupled from (p,φ). 4.2 Flow with small viscosity and conductivity In addition to the unrealistic case Pr θ = 1 that permits simultaneous factorization of L and source terms, another case where the flow exhibits viscous hyperbolic wave governed by lower-order equations is when the viscosity and heat conductivity are small: ν, ν θ, k,κ = O(δ), δ 1. (22) Here, to keep generality we do not compare the magnitude of δ and ɛ. The second term of Eq. (15b)isnowofO(δ 2 ) and negligible. We shall see that theory under this assumption has quite different property from and much wider practical applications than that with Pr θ = 1. The problem amounts to whether or under what conditions can the source terms of different variables be separable. Depending on the magnitude of heat addition, we consider two subclasses Flow with small heat addition Q = O(ɛδ) By Eq. (2b), the source-free linear entropy equation reads ρ 0 T 0 t s = k 2 T + ρ 0 Q, or T 0 t s = κ γ 2 h + Q, (23) and hence s = O(ɛδ). This is the case if the entropy is generated by the linear ( p, φ) disturbances to the otherwise still fluid, and/or if there is an external weak heat addition Q = O(ɛδ). TheninEq.(15c) the operator D κ/γ can be replaced by t because γ κ 2 t s = O(ɛδ 2 ), while the other quantities are all of O(ɛ) or O(ɛδ). This makes s be easily eliminated from Eq. (14) and it follows that [ Dβ 2 ] [ ] [ ] p m + Q =. (24) φ χ c 2 D θ Thus, after multiplying the adjoint of the operator matrix from the left, one obtains W b p = D θ (m + Q) 2 χ, W b φ = c 2 (m + Q) + D β χ, where W b = D θ D β c 2 2 = 2 t c 2 2 b t 2 + βν θ 4 = t 2 (c 2 + b t ) 2 + O(δ 2 ) t 2 (c 2 + b t ) 2 (25a) (25b) (25c) with b β + ν θ, as defined by Eq. (13). The homogeneous version of Eq. (25b) (m =χ = Q = 0), i.e. W b φ = 0, is in agreement with the linearized version of the nonlinear equation for φ derived by Kuznetsov in 1971 under the assumption (22)[12]. However, it should be noted that in Ref. [12] it was assumed that ν, ν θ, k,κ = O(ɛ) as well, and thus ν θ φ = O(ɛ 2 ). This would make Eq. (25) not a self-consistent system. To avoid this problem one may either extend the present theory to the nonlinear regime containing all O(ɛ 2 ) terms (which is beyond our present concern), or as we do here assume Eq. (22) instead, with δ ɛ, say δ ɛ 1/m, m > 2, such that terms of O(δ 2 ɛ) are negligible. In fact, for this small Q case there is a much simper derivation for the φ-equation. Multiply the first line of Eq. (17a) with a factor of c 2, and minus the second line after being operated by t,wehave (c 2 2 D θ t )φ c 2 t s = c 2 m t χ, (26) then by substituting the third line of Eq. (17a) to eliminate t s, one arrives at ( ) κ (c 2 2 D θ t )φ c 2 γ 2 h + Q = c 2 m t χ. (27) Then using the second line of Eq. (17a) again to eliminate h, there is (c 2 2 D θ t )φ ( κ c 2 γ 2 s κ D θ 2 φ+ κ ) 2 χ + Q γ h 0 γ h 0 = c 2 m t χ

8 362 F. Mao et al. or W b φ = c 2 (m + Q) + D β χ κ γ c2 2 s, (28) in consistency with Eq. (25b). Evidently, the dilatation ϑ = 2 φ is also a permissible measure of the compressing process. Then so must be t ρ due to the linearized continuity equation t ρ = ϑ + m,for which one easily obtains W b t ρ = D β ( D θ m 2 χ)+ c 2 2 Q. (29) Finally, let us consider the equation for s. Equation (14) gives D κ/γ s = β 2 p + Q, which is now simplified to s s = β 2 p + Q. (30) Applying W b to both sides, using Eq. (25a), and dropping O(ɛδ 2 ) terms, we obtain ( t 2 c2 2) ( t s =β D θ 2 m β 4 χ + t 2 c2 2) Q. (31) Thus, t s is governed by the classic inviscid wave equation. Notice that this lower-order decoupled entropy equation does not come from any operator/source factorization but from the assumed smallness of s and Q. This is a signal that larger Q will make the order of entropy equation no longer reducible Flow with large heat addition Q = O(ɛ) This is the general case, which happens when the entropy is generated not only by the viscous acoustic mode itself or additional weak heat Q = O(ɛδ), but also has a stronger heat source Q = O(ɛ). In this case there is s = O(ɛ) as well and no variable can be eliminated apriori. We now generalize this concept to the case with Pr θ = 1 and non-vanishing sources. We split the entropy s into the sum of acoustically generated entropy s ac = O(ɛδ) and externally generated entropy s en = O(ɛ) (say, by a heat addition Q = O(ɛ)), with s en = 0ifQ = 0. We then decompose the field into acoustic mode t 2 t c 2 D θ 0 β 2 0 D κ/γ p ac φ ac s ac m+κ/γ 2 s ac = χ, κ/γ 2 s ac (32a) and entropy mode t 2 t p en κ/γ 2 s ac c 2 D θ 0 φ en = 0. β 2 0 D κ/γ s en Q+κ/γ 2 s ac (32b) Thus, the compressing process ( p, φ, s) governed by Eq. (17a) with given m,χ,q may be expressed uniquely as the sum of an acoustic mode (p ac,φ ac, s ac ) with Q = 0 and s ac = O(ɛδ), and an entropy mode (p en,φ en, s en ) with (m,χ,q) = O(ɛδ, 0,ɛ). Here, (p ac,φ ac, s ac ) satisfy W b p ac = D θ m 2 χ, W b φ ac = c 2 m + D β χ, t s ac = β 2 p ac. (33a) (33b) (33c) The equations for (p ac,φ ac ) remain the same as Eq. (25) with Q = 0, while that for s implies that the entropy is a byproduct of viscous pressure wave. The lower-order decoupled equation for s ac = O(ɛδ) is still Eq. (31) with Q = 0. In contrast, p en,φ en, s en and ρ en satisfy Lp en = t D θ Q, Ls en = W θ Q, (34a) Lφ en = c 2 t Q, Lρ en = c 2 2 Q. (34b) Thus, when Pr θ = 1, L can no longer be factorized; the acoustic and entropy modes are closely coupled even for linearized flow with small viscosity and heat conductivity. This situation has been anticipated the in preceding subsection. Physically, once a fluctuating heat source Q is of O(ɛ), it alone can generate a fluid motion with all compressing variables of the same order (e.g., conceive the spherically symmetric motion produced by a point source of heat). When Pr θ = 1, we have L = W κ D θ and the equation for p en is reduced to W κ p en = t Q, in consistency with Eq. (12). Thus, the same is true for p ac + p en which was defined by Chu and Kovasznay [6] as the acoustic pressure. The preceding analysis can be summarized as follows. Under the same condition of Splitting Theorem and for fluids with viscosity and conductivity of O(δ) 1, the operator L can be factorized and the acoustic mode exists, satisfying W b f = S f, (35) where f can be any one of p, φ, ϑ or t ρ, and S f is a corresponding source term depending on the choice of f. Moreover: 1. When Q = O(ɛδ), only the acoustic mode is an independent subprocess. The entropy is a byproduct of the acoustic mode and its time derivative obeys the classic wave equation with inhomogeneous source terms. 2. When Q = O(ɛ), in addition to the acoustic mode there is a separate entropy mode: W b f ac = S ac, L f en = S en, (36) where f ac + f en = f and S ac + S en = S f, but in the latter all compressing variables are strongly coupled. It should be remarked that, in the preceding Sects and the operator factorizations for the unbounded fluid were studied separately for Q = O(ɛδ) and Q = O(ɛ),

9 On a general theory for compressing process and aeroacoustics 363 respectively, which was just for clarity. In reality these two cases may coexist as the far- and near-field behavior of the same disturbance flow. Namely, the near field has Q = O(ɛ) with an acoustic mode and an independent entropy mode, where variables in the latter are strongly coupled; while the far field is reasonably described by an acoustic mode with entropy wave and Q = O(ɛδ) as a by-product, like in most modern acoustic theories such as Lighthill s. Here, the decomposition of compressing process in the far field depends not only on the operator factorization but also on the boundary condition which in general couples the acoustic mode and entropy mode. But, if we set the boundary separating the near source and far-field region sufficiently far away from the heat source, the values of pressure and entropy at the boundary caused by the heat source is negligible. So the operator factorization employed in Sect is applicable outside that boundary. This observation indicates that the present theory with arbitrary sources, although linearized, applies to both regions. 5 Concluding remarks By using the operator factorization method, the existing linear theories for polytropic gas in unbounded domain with arbitrary sources m, f, and Q has been systematically revisited. The work confirms the assertion of previous studies (Splitting Theorem) that the splitting of linearized flow into longitudinal process and transversal process is always applicable and free from ambiguity. As one is focused on the compressing process, our study leads to a unification, modification and generalization of the previous work. The main new results and their physical implications can be summarized as: 1. In general, the linear compressing process can be fully described by a single independent variable with other variables being dependent and derivable from that single independent variable. The most primary independent variable is the kinematic velocity potential φ, of which the solution alone can lead to the full description of the entire compressing process. 2. The governing equations for all variables have the same homogeneous high-order linear operator but different inhomogeneous source terms. This general compressing process can not be split into submodes without further approximation or assumptions. 3. The compressing process can be split into decoupled acoustic and entropy modes if either Pr θ = 1orthe viscosity and conductivity are small or negligible. For the former only p is qualified to be the general acoustic variable governed by W κ p = S p ; while the latter is the case encountered in practice, where any of p, φ, ϑ, or t ρ alone can describe the acoustic mode governed by W b f = S f ( f stands for any of these variables). 4. For the case of small viscosity and conductivity with Pr θ = 1, when the heat source Q is small, the acoustic mode dominates and entropy mode is a byproduct. But when Q is large enough, the acoustic mode as above coexists with an independent entropy mode, and in the latter all variables are no longer decoupled. These two cases may coexist in the same disturbance flow, where near the source there is Q = O(ɛ) but in the far field Q reduces to be of O(ɛδ). The decomposition of linearized compressing process carried out in the physical space as made in this paper gives an instinctive description of the evolution of acoustic mode and entropy mode. For those cases that the acoustic mode and entropy mode can not be decoupled, the analysis of the linearized equation in spectral space is also useful, where the coupling and interaction of different modes can be formulated by the ordinary equations system. For the far-field fluid which ignores the boundary the analysis in the spectral space is also applicable. In fully nonlinear regime, the compressing process will be a generalization of the acoustic mode coupled with the entropy process as well as the shearing process. In this case and for arbitrary Pr θ but small viscosity and conductivity, if the shear flow field and entropy field are given, then by the above results of linear theory one expects that the most compact formulation for compressing process is in terms of enthalpy h that governs the variation of sound speed and any one of the above permissible compressing variables. 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