An Ontological Basis for Describing Combustion Chamber Oscillations

Abstract

The Temporal Theory of High-Frequency Instability (TTHI) offers a unified ontological basis for describing oscillations in combustion chambers. In this model, time is treated as a physical substance with a local rate  = d/dt and a potential  = ln ; spatial gradients  generate inertial force densities f_T = c' . A phenomenological relation  = (T, p, s) is introduced, demonstrating the mathematical equivalence between "temporal energy pumping" and the classical Rayleigh criterion  p(t) q(t) dt > 0. The theory does not replace thermoacoustics but deepens it by revealing the primary cause of the phase-based energy supply to modes. Falsifiable effects are predicted: splitting of degenerate transverse modes under a radial gradient (r) and a frequency shift of longitudinal modes under an axial gradient /x. For practical application, a control metric R = ()"u (averaged over period P) is introduced, and strategies for suppressing instability through smoothing temporal gradients are proposed, paving the way for more stable and efficient energy systems.

Keywords

Temporal Theory of High-Frequency Instability (TTHI); Temporal Theory of Gravity (TTG); time as a substance; rate of time  = d/dt; temporal potential ; temporal gradient ; temporal force f_T = c' ; thermoacoustic instability; Rayleigh criterion; combustion chamber (LRE); acoustic modes; mode splitting; frequency shift; instability control; R metric; phenomenological model.

Table of Contents.

1. Introduction
   1.1. Motivation: HFI in Combustion Chambers and the Limits of Classical Thermoacoustics
   1.2. The TTHI Idea: Time as a Substance, Rate , Potential
   1.3. Contribution and Article Structure
2. Ontological and Mathematical Basis of TTHI
   2.1. Definitions: = d/dt, = ln
   2.2. Temporal Force: f_T = c'
   2.3. Constitutive Relation with Thermodynamics: (T, p, s) - a_s"s + a_T"ln T + a_p"ln p
   2.4. Scale Estimates of || for LREs
3. Connection to Classical Thermoacoustics
   3.1. Rayleigh Criterion: p(t) q(t) dt > 0
   3.2. Equivalence of Energy Pumping: W c' ()"u(t) dt p(t) q(t) dt > 0
   3.3. Interpretation of Phasing and Delays in the -Language
4. Eigenmodes, Resonance, and Growth Mechanisms
   4.1. Chamber+Nozzle Geometry: Longitudinal/Transverse Modes
   4.2. Resonance at Coincidence of _puls() and _mode
   4.3. Modal Energy Balance in Terms of
5. Falsifiable Predictions of TTHI
   5.1. Radial Gradient (r) Splitting of Degenerate Transverse Modes (Fig. G)
   5.2. Axial Gradient /x Frequency Shift of Longitudinal Modes (Fig. H)
   5.3. Sign and Magnitude of f, split as Functions of /x, |/r|
6. Diagnostics and Experiment
   6.1. Proxy Metrics: Schlieren/Shimmer (refractive index), PLIF/CH* for q, Fast p Sensors, PIV/LDV for u
   6.2. How to Measure R_ = ()"u_P
   6.3. Verification Protocols: Imposed /x and /r, Stability Maps
7. Control and Instability Suppression
   7.1. Target Metric: R_ T 0
   7.2. Control Levers: q(t) Phase Distribution (Staggering), Wall Cooling/Heating (Smoothing ||), Pilots, "Temporal Resistors"
   7.3. Integration with Resonators/Dampers (Classical + TT-View)
8. Applicability Limits and Discussion
   8.1. Where TTHI Adds Value and Where It Is Equivalent to Classical Theory
   8.2. Sensitivity to Choice of a_s, a_T, a_p and Calibration
   8.3. Potential Interpretation Errors (Not "Energy from Nowhere")
9. Conclusion
   Summary, Practical Conclusions, Future Work (Metamaterials/Distributed Heating/Active R_ Control).
10. References

Appendices
A. Derivation of Pumping Equivalence (Rayleigh -Formalism)
B. Pseudographic Schemes (Monospace Font)
C. Table of Symbols and Dimensions (, , f_T, R_, q, p, u, a_s, a_T, a_p)
D. Order-of-Magnitude Estimates for Prototype LREs
E. Experimental Protocol Template (Sensors, Sampling Rates, Phase Synchronization)

1. Introduction

1.1. Motivation: HFI in Combustion Chambers and the Limits of Classical Thermoacoustics

High-Frequency Instability (HFI) in liquid rocket engine (LRE) combustion chambers remains one of the most challenging problems in engine design. The classical framework of non-equilibrium thermodynamics and acousticsincluding the Rayleigh criterion and modal analysis of the chamber+nozzle systemsuccessfully describes the conditions for instability onset (phasing of pressure p and heat release q oscillations) but does not reveal the primary cause of the phase-based energy supply to modes: why does the system tend to maintain the required phase lag, and where is the universality of this mechanism encoded? This creates a methodological limit: suppression reduces to dampers, injector reconfiguration, tuning of empirical parameters, and "detuning" phases, while fundamentally new strategies remain weakly justified.

Engineering Context. For typical chambers: eigenfrequencies range from several kHz to tens of kHz; pressure oscillations are on the order of 0.11 MPa; characteristic scales are tens of centimeters. Even small phase-based energy inputs per period can trigger an avalanche-like growth of amplitudes.

1.2. The TTHI Idea: Time as a Substance, Rate , Potential

A paradigm stemming from the Temporal Theory of Gravity (TTG) is proposed, where time is considered not as a purely geometric parameter but as a physical substance with properties at each point in the flow.

• Rate of Time:  = d/dt.
• Temporal Potential:  = ln .
• Temporal Force (Force Density): f_T = c' .

The key idea of TTHI: oscillatory processes in the chamber are the dynamics of temporal gradients. Inhomogeneous energy release forms regions of different "temporal density" (different and thus ). At the boundaries of such regions, gradients appear, equivalent to inertial "phase pressures" capable of pumping energy into acoustic modes. In this description, resonance occurs when the frequencies of pulsations fall within the vicinity of the chamber's eigenfrequencies.

Connection to Classical Theory. To "stitch" TTHI with measurable quantities, a phenomenological relation = (T, p, s) is introduced. Then is expressed through conventional thermodynamic gradients, and the mode growth criterion is formulated as equivalent to Rayleigh's, not breaking the classical picture but deepening it.

1.3. Contribution and Article Structure

Main Results and Contribution:

1. TTHI Mathematical Framework. Variables , and the force term f_T = c' are introduced; the phenomenological relation = (T, p, s) is defined, allowing expression of through measurable temperature, pressure, and entropy fields.
2. Pumping Equivalence. It is shown that the temporal energy pumping into modes is written as c' ()"u dt > 0 and is mathematically equivalent to the Rayleigh criterion p(t) q(t) dt > 0, i.e., TTHI deepens, rather than replaces, the thermoacoustic paradigm.
3. Falsifiable Predictions. Measurable effects are predicted:
   Splitting of degenerate transverse modes under a radial gradient (r);
   Systematic frequency shift of longitudinal modes under an axial gradient /x.
   The scales of these effects are linearly related (to first order) to the averaged values |/r| and /x.
4. Metrics and Control. The control metric R = ()"u_P (averaged over period P) is introduced as a target for instability suppression design; strategies for smoothing temporal gradients (distributed heating/cooling, q phase staggering, "temporal resistors") are proposed.

Article Structure:

• Section 2. Basic TTHI equations and the phenomenological relation (T, p, s); scales and estimates.
• Section 3. Equivalence of the temporal formalism to the Rayleigh criterion; modal energy balance.
• Section 4. Testable predictions (splitting, shift, stability maps).
• Section 5. Engineering applications: control metrics (R), methods for smoothing , integration with classical dampers.
• Section 6. Conclusion and outlook: test stand verification, calibration of (T, p, s), extension to solid/hybrid systems.

Glossary, Chapter 1.
- rate of time; - proper time; t - laboratory time; - temporal potential; - gradient operator; - density; c - speed of light; u - oscillatory velocity; p - oscillatory pressure; q - oscillatory heat release; R - control metric ()"u over period.

2. Ontological and Mathematical Basis of TTHI

2.1. Definitions: = d/dt, = ln

Within the temporal ontology, time is treated as a physical substance with local properties.
 Rate of Time:  = d/dt  the ratio of the system's proper time to laboratory time.
 Temporal Potential:  = ln   a dimensionless scalar field quantity.

Notes:
is dimensionless; small variations are convenient in the form  -  for || 1.
The transition "time-as-parameter time-as-substance" makes the appearance of field gradients and associated force effects admissible.

Figure 1. A Field and vectors (qualitative)

'?????????????? chamber ????????????????

? ?    ? ? center is hot is higher

? ? ? ? near walls cooler is lower

? "    ў ? points from fast zones to slow

? ?

? ?

"?????????????????????????????????????...

2.2. Temporal Force: f_T = c'

In TTHI, spatial variations in the rate of time impose an effective volumetric force of inertial nature:
(1) f_T = c' ,
where is the medium density, c is the speed of light, is the gradient of the temporal potential. This force enters the local momentum balance as an additional "body" term (dimensionally N/m) and serves as the universal carrier of phase-based energy pumping into modes. The energy contribution over period P:
(2) W_T f_T " u(t) dt = c' () " u(t) dt.
This expression will be used in 3 to connect with the Rayleigh criterion.

Figure 2. B TTHI "Force Chain"

text

T, p, s (Thermo-fields)

?

(Phenomenology: = a_s"s + a_T"ln T + a_p"ln p)

(Temporal Potential)

?

(Spatial Gradient)

(Temporal Gradient)

?

(Temporal Force per Volume)

f_T = c' W_T c' ()"u dt

2.3. Constitutive Relation with Thermodynamics: (T, p, s)

To "stitch" TTHI with measurable quantities, we introduce a phenomenological relation:
(3) = (T, p, s) - a_s " s + a_T " ln T + a_p " ln p,
where T is temperature, p is pressure, s is specific entropy; a_s, a_T, a_p are calibratable coefficients (dimensionless in this form). Then
(4) - a_s s + a_T (T / T) + a_p (p / p).

Note (approximately ideal gas):
for s - c_p ln T R ln p + const we get an equivalent two-parameter form
(5) - A_T ln T + A_p ln p, where A_T = a_T + a_s c_p, A_p = a_p a_s R. This is convenient for calibration against test data.

2.4. Scale Estimates || for LREs

We provide two independent estimates for the order of magnitude of ||.

(A) From required energy pumping into a mode.
In a steadily growing HFI, the characteristic scale of the "pumping" force per unit volume is on the order of p / L, where p is the pressure oscillation amplitude, L is the characteristic chamber length. Equating this to |f_T| = c' ||, we get
(6) || - (p) / ( c' L).
Substituting typical numbers for LRE chambers (e.g., p - 0.5 MPa; - 1 kg/m; L - 0.5 m) gives
|| ~ 10 m.
This shows that to reproduce observed pumping levels, very small temporal gradients are sufficientconsistent with compactly parameterizing ordinary thermo-gradients.

(B) Via measurable thermo-gradients.
From 2.3:
(7) || |A_T| " |T|/T + |A_p| " |p|/p.
For characteristic chamber conditions (|T|/T ~ 10 m, |p|/p ~ 10'...10 m, coefficients A_T, A_p calibratable numbers of order 10'...1), the typical first order for || also gives 10'...10 m, compatible with estimate (A). The exact value is determined by calibrating A_T, A_p for a specific setup.

Figure 3. C Estimate of || (Path A)

text

Required "Pumping" Comparison with f_T Result

p ~ 0.5 MPa \|f_T\| - c' \|\| \|\| - p / ( c' L)

L ~ 0.5 m, ~ 1 kg/m, c ~ 3"10 m/s \|\| ~ 10 m

2.5. Linearization and Connection to the Rayleigh Criterion (Preview)

For small perturbations  = + , u = u + u, the energy pumping into a mode per period:
(8) W_T c' () " u dt.
Considering (T, p, s), we obtain equivalence to the Rayleigh criterion in standard variables: the sign of W_T coincides with the sign of  p(t) q(t) dt. That is, the temporal formulation does not contradict classical theory but makes explicit the primary cause of phasing: mode growth is due to the in-phase relationship of oscillations of  and u (which corresponds to p and q in thermo-variables).

Mini-Glossary of Symbols (for Section 2)
- rate of time; - proper time; t - laboratory time; - temporal potential; - gradient; - density; c - speed of light; u - oscillatory velocity; p - oscillatory pressure; q - oscillatory heat release; a_s, a_T, a_p - calibration coefficients; A_T, A_p - their reduced combinations.

3. Connection to Classical Thermoacoustics

3.1. Rayleigh Criterion: p(t) q(t) dt > 0

The classical Rayleigh criterion states: a mode grows if the average work of heat release oscillations q(t) on pressure oscillations p(t) over period P is positive:
(9) p(t) q(t) dt > 0.
Intuitively: when heat release occurs at the "right" moments (in phase with pressure increase), energy flows from the chemical source into the acoustic mode. This criterion is phenomenological but agrees well with experiments and calculations.

Classical Assumptions (Linear Theory):
Small perturbations: p = p + p, T = T + T,  = + , u = u + u, |X| |X|;
Weak mean mass flow in the modal analysis zone (or accounting for it as a small correction);
Rigid walls in the first approximation;
Relation between p, , u via acoustics and thermochemistry; q is the source in the energy equation.

Figure 4. A Phasing of p and q (Growth per Rayleigh)

text

p(t): ??/\??/\??/\??

q(t): ?/\??/\??/\?

In-phase p q dt > 0 Growth

3.2. Equivalence of Pumping: W c' ()"u(t) dt p(t) q(t) dt > 0

In TTHI, the growth of modal energy is driven by the temporal force density
(10) f_T = c' .
The work of this force over period P:
(11) W_T f_T " u(t) dt = c' ()"u(t) dt.
Below is a "sketch" of how this expression reduces to the Rayleigh criterion under the phenomenological relation  = (T, p, s).

Derivation Sketch (Linearization, First Order):

1. Integration by parts (for fixed, weakly permeable walls; surface terms are small):
   (12) ()"u dt = ("u) dt.
2. Mass continuity (linear):
   (13) /t + ("u) - 0 "u - (1/) /t,
   therefore (14) ()"u dt - (1/) (/t) dt.
3. Relation between , p and q (ideal gas as a guide): without heat source q = 0  quasi-isentropically p - c_s' ; with source q, phase lags between p and  are determined by the energy equation, where q is an addition to the enthalpy/entropy balance.
   In TTHI phenomenology: (15) - a_s s + a_T (T/T) + a_p (p/p).
4. Substituting  and using the energy equation leads to sign coincidence:
   (16) c' ()"u dt > 0 p(t) q(t) dt > 0.

Meaning. "Temporal pumping" is not different physics but a compact parameterization of the same phase-based energy supply condition: instead of "p and q are in phase," we say "fluctuations of are in phase with u."

Limitations of Equivalence:
Linear regime (small perturbations);
Correct calibration of a_s, a_T, a_p for composition and regime;
Accounting for boundary conditions (leaks/active elements give surface contributions);
For strong mean flows account for convection and mode drift.

Figure 5. B Equivalence in -Language (Sign of Work)

text

:

u:

()"u: R = ()"u_P < 0

W_T c' R > 0 Mode Growth (Equivalent to Rayleigh)

3.3. Interpretation of Phasing and Delays in the -Language

What "Phase" Means in TTHI.
 In-Phase Growth. A mode grows when, in zones of intense heat release, the oscillatory components  and u give a negative contribution to ()"u (due to the overall "" in the formula for W_T), i.e., when  c' ()"u dt > 0. In classical language, this corresponds to  p q dt > 0.
 Role of Delays. Delays in chemical kinetics, evaporation, mixture formation, as well as "acoustic" delays (injectors, throat) manifest in the -language as inertia in the restructuring of the temporal structure:  responds to T, p, s with a phase lag. When the lag is such that the pair (, u) is "in the right phase," the mode receives energy.

Resonance: Frequency Coincidence.
 Condition: The spectrum of  pulsations contains components close to the chamber's eigenfrequencies (longitudinal, transverse modes).
 Consequence: The corresponding modes are pumped analogous to the spectrum of q falling within modal frequencies in the classical theory.

Practical Reformulation of Control.
Instead of the abstract "control the phase of p and q," we set the goal to control the sign and magnitude of
(17) R = ()"u_P, aiming for R T 0.
This is achieved by levers that deliberately shift (T, p, s) into a "detrimental" phase for the modes: distributed wall heating/cooling, q phase distribution across injectors (staggering), local pilots, "temporal resistors" (analogous to resonators but targeting the phase of ).

Figure 6. C Phase Diagram: Lag(u) and Growth/Decay

text

Lag( u)

'?????????? Growth ??????????? Decay

0R ? 30R 60R 90R 120R ? 180R

c'()"u > 0 ? < 0

Amplitude

(Reminder: Growth **R < 0**, since **W_T c' R**.)

3.4. Correspondence Table: Classical TTHI (for Quick Reference)

3.5. Applicability Conditions and What to Do if "It Doesn't Match"

 Strong Mean Flows/Shear Layers: Add convective terms and mode drift; account for inhomogeneities (-form carried over to equations including u).
 Permeable Boundaries/Resonators: Surface terms in integration by parts can be significant this gives "boundary pumping"; in the -language, it corresponds to an imposed structure at the boundary.
 Strong Nonlinearity: At large amplitudes, harmonics of and subharmonics appear; the Rayleigh equivalence remains a guide for the first approximation, but extension to weakly nonlinear terms is required.

Section 3 Summary

1. TTHI recovers the Rayleigh criterion, making explicit the primary cause of phase-based pumping via .
2. Instability control can be formulated as achieving R T 0  disrupting the in-phase relationship of the pair (, u).
3. TTHI predictions (splitting of transverse modes under (r), frequency shift under /x) are natural consequences of the same "phase-based pumping" but in a unified temporal parameterization.

4. Eigenmodes, Resonance, and Growth Mechanisms

4.1. Chamber + Nozzle Geometry: Longitudinal and Transverse Modes

Modal Picture. Acoustic modes of "chamber + throat + nozzle" are eigen-solutions of the wave equation with boundary conditions on the injector face, walls, throat, and further in the nozzle tract (with partial reflection):

• Longitudinal (Axial) Modes. Pressure/velocity nodes/antinodes stretch along the axis; in TTHI, they correspond to stationary patterns (x) and (x).
• Transverse (Radial/Tangential) Modes. Annular/azimuthal structures of (r,) form in the cross-section; degenerate modes (e.g., a pair of modes with the same frequency) are sensitive to radial asymmetry.

Boundary Conditions in Terms of Impedance. An effective acoustic impedance Z() is specified at the injector face and throat, determining the reflection coefficient. In the -language, this is equivalent to an "imposed" phase/amplitude of at the boundary (via (T,p,s)).

Frequency Estimates (First Approximation):

• Longitudinal: f - n"c_eff/(2L_eff), n = 1,2,...
• Radial: f_(m,0) - _m"c_eff/(2R), _m roots of Bessel functions.
  Here c_eff is the effective speed of sound (accounts for mean T,p), L_eff, R are effective sizes accounting for the throat and nozzle bell.

Scheme 1. (Axial mode in chamber with nozzle):

text

Injectors Throat Nozzle

'?????????? Chamber ??????????? '???? '???????????????

? p p p ? ? ? ? Radiation

? u u u ? ? ? ? (Partial)

"????????????????????????????... "???...

-pattern: ---_______------_______--- (nodes/antinodes along axis)

(x): (changes sign between antinodes)

Scheme 2. (Transverse mode, flow view):

text

'???????????? r, ?????????????

? ?? ?? ? Tangential Pair (Degenerate)

? ???? ???? ? (r,) forms "lobes"

? ?? ?? ? Sensitive to wall/heating asymmetry

"??????????????????????????????...

4.2. Resonance at Coincidence of _puls() and _mode

In TTHI, amplitude growth is caused by the spectral components of  pulsations falling within the vicinity of the mode's eigenfrequency:
(18) _puls() - _mode.
Here _puls() is determined by the unsteadiness of combustion (chemical kinetics, evaporation, mixture formation, heat release oscillations q). When coincidence occurs, effective energy transfer from the "temporal field" to acoustics takes place via the work of the force f_T = c' .

Role of Phasing. For growth, the correct phase lag between and u is required (see 3): when averaged over a period, R = ()"u_P must give a negative contribution to the pumping formula W_T c' R (i.e., c' R > 0).

Scheme 3. (Classical resonance curve, interpreted in -language):

text

Amplitude

/|\

/ | \ /\ Detuning = _puls() _mode

/ | \_/ \_\

_____/___|___\_|___________|_\_____

_mode

At _puls() - _mode and "right phase" (R < 0) amplitude grows.

4.3. Modal Energy Balance in Terms of

Introducing the mode energy E_mode and dissipative losses D (viscosity, thermal conductivity, radiation into the nozzle, wall losses), we obtain the balance:
(19) dE_mode/dt = c' ()"u_P D.

• Temporal Pumping: P_T = c' ()"u_P.
• Amplitude Growth: when P_T > D.

Connection to Growth Rate . For a mode with frequency and equivalent "kinetic" energy E_mode - (1/2) M_eff u' (M_eff effective modal mass), we have, to first order:
(20) - (P_T D) / (2 E_mode).
Where P_T is expressed via R:
(21) P_T = c' V_eff " R,
V_eff is the effective modal participation volume. By controlling R, we directly control .

Scheme 4. (Energy Flows):

text

Chemical Energy q(t) (T, p, s) u E_mode

Losses D (Visc., Walls, Radiation)

Growth Condition: c' ()"u_P > D

4.4. Predicted TTHI Effects (Connection to Geometry and Gradients)

(a) Splitting of Transverse Modes under Radial Gradient (r).
Radial asymmetry (e.g., hot center cold walls) breaks the degeneracy of transverse modes and leads to frequency splitting: f_(1,0) f_(1,0)+ and f_(1,0). To first order, f |/r| over the cross-section.

Pseudographics (Flow View):

text

'???????????? Chamber ?????????????

? ??? """" """" ??? ? (r) (Radial)

? ?"" """" """" "? ? Degenerate Pair Splitting

"????????????????????????????????...

(b) Frequency Shift of Longitudinal Modes under Axial Gradient /x.
Progressive heating/cooling along the axis creates /x 0, changing the "stiffness" of the medium. To first order: f /x along L_eff. The sign of the shift is determined by the sign of the gradient.

Pseudographics (Axial Profile):

text

(x): ?????? (Increasing towards nozzle) /x > 0 f

x "??????????????

4.5. Nonlinear Saturation and Practical Conclusions

• Nonlinear Effects. As amplitude grows, mean T,p change (hence and ), harmonics of appear; growth slows and reaches a stationary level (balance of P_T and D).
• Practical Control Metric. Target R = ()"u_P (C) 0. Achieved by:
  q phase distribution across injectors (staggering);
  Distributed wall heating/cooling (smoothing ||);
  "Temporal resistors" (analog of Helmholtz resonators targeting phase);
  Modifying impedance at the throat/injector face.

Section 4 Summary

1. Chamber geometry and impedance determine the mode spectrum; in TTHI, modes are stable patterns of and .
2. Resonance and amplitude growth occur at coincidence of _puls() with _mode under the "right phase" (R < 0).
3. Modal energy balance in -language: dE/dt = c' ()"u D; control reduces to achieving R T 0.
4. Falsifiable effects: splitting of transverse modes under (r), shift of longitudinal frequencies under /x scales are directly related to averaged gradients.

5. Falsifiable Predictions of TTHI

5.1. Radial Gradient (r) Splitting of Degenerate Transverse Modes (Fig. G)

TTHI predicts that the presence of a radial gradient of the temporal potential (r) lifts the degeneracy of transverse acoustic modes. Specifically:
Degenerate modes like (1,0) split into two modes with frequencies  and ;
The splitting magnitude _split is proportional to the averaged radial gradient: _split |/r|;
The sign of the "shift" for each branch is determined by the direction of the (r) gradient (hot center cold walls or vice versa).

Experimental Verification. The effect is measured by high-precision spectral analysis of transverse oscillations under controlled radial non-uniformity of wall heating/cooling.

Figure G Splitting of Transverse Modes under Radial (r)
(Flow view; monospace font)

text

Chamber (Flow View)

z-axis

?

'????????No?????????

? ???? ? ???? ? Colder at walls lower

? ?""" ? """? ? Radial Gradient (r)

? ?"""" ? """"? ?

? ?"""" No """"? ? Nodes/Antinodes of transverse modes

? ?"""" ? """"? ? shift under influence

? ?""" ? """? ?

? ???? ? ???? ?

"????????No????????...

?

??? r ???

Symmetry Breaking Frequency Splitting of Degenerate Pair:

f_(1,0) f_(1,0)+ and f_(1,0), f |/r|.

Important: Classical acoustics also predicts splitting under radial property gradients (T, , c). TTHI provides a compact parameterization via (T,p,s): the splitting magnitude should linearly depend (to first order) on the averaged /r.

5.2. Axial Gradient /x Frequency Shift of Longitudinal Modes (Fig. H)

An axial gradient of the temporal potential /x causes a systematic shift of longitudinal mode frequencies:
All longitudinal frequencies change by _shift;
The sign and scale of the shift are determined by the axial mean gradient: _shift /x;
The effect is stronger for lower longitudinal modes and in some regimes can reach 15 % of the nominal frequency (first-order estimate).

Experimental Verification. Measurement of frequency responses under imposed temperature profiles along the axis (changing heat flux/cooling distribution) and comparison with calculated /x from (T, p, s).

Figure H Frequency Shift under Axial Gradient /x

text

Axial Profile (x)

(Rate of Time)

? ????? (x) increases towards nozzle

? ???? /x > 0

? ????

?????

"?????????????????????????????????????? x

Injectors Nozzle

Without Gradient: ?? (Frequency f)

With Gradient: ? ? (Frequency f > f for /x > 0)

Estimate: f /x (along chamber length).

5.3. Sign and Magnitude of f, split as Functions of /x, |/r|

TTHI establishes quantitative relationships between temporal gradients and observed frequency shifts:

For Longitudinal Modes:
(22) f / f = k_x " /x + O(/x').

For Transverse Modes:
(23) split / f = k_r " |/r| + O(|/r|').

where:
 f is the original mode frequency without temporal gradients;
 k_x, k_r are calibration coefficients (dependent on chamber geometry, boundary conditions, and mixture composition);
 " denotes averaging along the corresponding direction (x along chamber, r over cross-section).

Practical Verification Procedure.

1. Impose controlled T(x) or T(r) profiles (via cooling/heating, pilots).
2. Compute /x, |/r| from (T, p, s).
3. Measure frequencies: f_mode(gradient) and f(reference).
4. Plot f/f and split/f as functions of average gradients; estimate k_x, k_r by regression. Agreement in sign and scale is direct verification of TTHI.

5.4. Additional Predictions (Testable)

 Stability Map in (Detuning, R) Coordinates. At fixed detuning  = _puls() _mode, the transition "growth decay" occurs when the sign of R = ()"u_P changes.
 Boundary Pumping. Changing the "temporal impedance" on the walls (imposed via local heating/cooling) yields an observable shift in the self-excitation threshold.
 Nonlinear Saturation. As amplitude grows, and change, leading to "energy transfer" between modes and the appearance of subharmonics in a predictable saturation pattern.

5.5. Applicability Range of Estimates

The given linear dependencies and proportionalities are valid to first order for small gradients || 10...10 m, weak nonlinearity, and correct calibration of (T, p, s). For regimes with strong mean flows and high amplitudes, convection, boundary terms, and weakly nonlinear corrections must be accounted for the methodology remains the same, but coefficients k_x, k_r are recalculated.

6. Diagnostics and Experimentation

6.1. Proxy Metrics for and Measurable Fields

To verify TTВН, a complementary set of methods is used to reconstruct and from thermodynamic fields and flow data:

• Schlieren/Shimmer (n as proxy for ): The refractive index n(x,t) is linked to and T. Using the phenomenological relation = a_s"s + a_T"ln T + a_p"ln p, we build a proxy for from n data (after converting to T/T and p/p) and calibrating a_s, a_T, a_p.
• PLIF and CH (mapping q(x,t)): Planar laser-induced fluorescence and chemiluminescence provide q(x,t) with high temporal and spatial resolution input for (T,p,s).
• Fast pressure sensors p(t): Sampling rate T 50200 kHz (based on highest mode, minimum 510 of f_max of interest).
• PIV/LDV (u): LDV pointwise high-frequency velocity u(t); PIV velocity field u(x,t) with lower frequency but spatial coverage.
• Thermocouples / IR pyrometry / thermal imaging (T, T): Mean profiles T(x,r) and gradients for calibrating (T,p,s).

Diagram 5.A Sensor Layout (Side View)

Код

Injectors C A M B E R Throat Nozzle

'??????????????? '?????????????????????????????????????? '?????? '??????????????

? PLIF / CH* ? ? p: p: p: ? ? ? ? Schlieren / ?

? (q map) ? ? LDV: PIV: IR: ? ? ? ? Shimmer port ?

"??????????????... ? TC: proxy-: n ? "?????... ? n(x,t) ?

? a_wall: : (from =T,p,s) ? "??????????????...

"?????????????????????????????????????...

optical window / laser sheet (PLIF/PIV) stinger (thrust)

Legend: p (fast sensor), LDV, PIV window, IR/thermography,

thermocouple/RTD, accelerometer (wall/nozzle), n Schlieren/Shimmer

Proxy Calibration (Static and Dynamic)

1. Measure T(x,r), p(x,r) in steady state
2. Choose a_s, a_T, a_p (see 2.3), estimate A_T = a_T + a_s"c_p, A_p = a_p a_s"R
3. Construct (x,r) - A_T"ln T + A_p"ln p, then
4. For fluctuations: - a_s"s + a_T"(T/T) + a_p"(p/p); T from PLIF/Shimmer (via n), p sensors, s from energy equation (enthalpy balance)

6.2. Measuring the Metric R = ()"u

Target metric for temporal energy pumping:

• Locally:R(x) = (1/P) " ()"u dt
• Volume-integrated:R = (1/V) " _V R(x) dV
• Suppression criterion:R T 0 (see 7.1)

Protocol

1. Synchronously record (x,t) and u(x,t) at matching points/cells
2. Compute S(x,t) = ()"u
3. Average over period P R(x)
4. Integrate over volume R

Resolution Requirements

• Time (C) 10 s (up to ~100 kHz)
• Space (C) 1 mm (in zones of peak )

Diagram 6.B Temporal Synchronization of Channels

Код

Trigger ??? p(t) ????????????????????????????????? (reference channel)

No? q(t) (PLIF/CH*) ????????????No???????? (optical frame marker)

No? u(t) (LDV/PIV) ?????????????No???????? (LDV clock / PIV markers)

"? Schlieren/Shimmer ????????????%???????? (lamp/shutter/modulated source)

All phases are aligned to p(t) from the reference sensor (via cross-correlation or markers).

Fast Approximate Protocol

If u is pointwise (LDV) and -proxy is field-based (Schlieren), take u at nodes/antinodes of target mode, from nearby regions, and compute R as average over representative points.

6.3. Verification Protocols (Imposed Gradients and Stability Maps)

(A) Axial Gradient /x Frequency Shift of Longitudinal Modes Method: stepped/gradient heating of chamber walls along axis Measurements: longitudinal mode frequencies f_mode; reference f (no gradient) Relation:f/f - k_x " /x (averaged over chamber length)

(B) Radial Gradient /r Splitting of Transverse Modes Method: asymmetric heating/cooling, offset injection Measurements: f_(1,0)+, f_(1,0);split = f_(1,0)+ f_(1,0) Relation:split/f - k_r " |/r|

(C) Stability Maps (DetuningR) X-axis: = _puls() _mode Y-axis:R

Diagram 7.C Stability Map in (, R) Coordinates

Код

R

growth

? """""""""

? " "

? " "

R=0 ?No???"????????????"????????

? ? ?

? ????????? decay: R T 0 or || 0

Verification Criterion: Measured f and split agree with TTВН predictions within 10% given calibrated a_s, a_T, a_p and proper error accounting.

6.4. Reconstruction of and Error Estimation

Reconstruct (x,t):

1. Measure T(x,t), p(x,t); if possible, estimate s(x,t) (enthalpy balance)
2. Compute (x,t) - a_s"s + a_T"ln T + a_p"ln p
3. Obtain (x,t) via numerical differentiation (with regularization to suppress noise)

Uncertainty Budget (First Order):

• from T/T and p/p: - |A_T|"(T/T) + |A_p|"(p/p)
• ():error from numerical gradient ~ O(h), where h is grid spacing
• R:dominated by phase misalignment and u noise aim for phase accuracy (C) 35R and SNR T 20 dB

6.5. Practical Recommendations

• Synchronization: unified hardware trigger; all optical channels phase-locked to p(t)
• Anti-aliasing: sampling rate T 510 f_max of interest
• Windowing/Averaging: Hann window; average T 50100 periods for stable R estimation
• Proxy Validation: compare computed from (T,p) with from Schlieren (via n) in steady state
• Reporting: alongside f, split, publish /x, |/r|, selected a_s, a_T, a_p, and error map

7. Control and Suppression of Instability

7.1. Target Metric: R T 0

Definition Local temporal pumping metric: R(x) = (1/P) " ((x,t) " u(x,t)) dt Integral metric over the chamber: R = (1/V) " _V R(x) dV

Energy relation Temporal pumping power over one period: P_T = " c' " R

Then:

• R < 0P_T > 0energy flows into the mode (pumping, growth)
• R = 0P_T = 0neutral stability
• R > 0P_T < 0energy is extracted from the mode (anti-pumping, decay)

Control goal for HFI suppression: achieve R T 0 (ideally, significantly positive)

Monitoring Practice

• Local monitoring: R(x) = (1/P) " ( " u) dt identifies hot spots of pumping (R(x) < 0) and anti-pumping (R(x) > 0)
• Integral control: R = (1/V) " _V R(x) dV target function for suppression tuning

Important note on sign: Ensure that the definition of u direction and axis orientation is consistent across all channels; otherwise, the sign of R may flip due to convention mismatches.

Diagram 7.A R Traffic Light (Control Ideology)

Код

R

+ ? anti-pumping stabilization

??????????????????????

0 ??No???????? neutral ????? goal: R T 0

?"""""""""""""""""""""

? pumping intervention

Mini Checklist for Test Stand

• Align channel phases to reference p(t) at selected point
• Verify sign: if amplitude growth is observed, data should yield R < 0
• For per-mode analysis, compute R_m = _V ("u)"_m dV / _V |_m| dV, where _m is the normalized velocity mode shape useful when modes overlap

7.2. Control Levers: staggering q(t), wall heating/cooling ( smoothing), pilots, temporal resistors

TTВН offers a set of control tools some modify phase, others affect || gradients, and some adjust impedance at boundaries.

Diagram 7.B Staggering q Across Injector Rings

Код

Injector rings (flow view):

[ A ] [ B ] [ C ] [ D ]

0R + 180R

Net in mode R 0 or < 0

(1) Phase Distribution of q(t) Across Injectors (Staggering) Goal: disrupt coherence between and u for the target mode

• Idea: shift heat release phases on injector groups by so the net visible for the mode becomes decorrelated from u
• Tuning: choose based on mode shape map (nodes/antinodes of u and )

Why?

• In azimuthal/radial modes, the main pumping comes from in-phase relative to u
• Sectoral staggering of q breaks this in-phase component, reducing net in the mode R = ()"u 0 T 0 (goal per Diagram 7.A)

Practice:

• Start with - 2040R between adjacent sectors; check mode spectrum (m=1,2) and tune for minimum |R|
• If needed, switch to 68 sectors (fine phase comb) to suppress multiple m simultaneously

(2) Wall Heating/Cooling (Smoothing ||) Goal: reduce radial/axial gradients of , and thus amplitudes of

• Radial: wall temperature equalization lowers |/r| reduces mode splitting and local pumping near walls
• Axial: shaping T(x) profile lowers /x reduces frequency shift and resonance sensitivity

Diagram 7.C Where to Smooth || Gradients

Код

Injector Chamber (wall heating/cooling) Throat Nozzle

"""""" """""" """"""

|(r)| |/x| impedance tuning at boundaries

How to read the diagram:

• Radial belt (chamber wall): local heating/cooling around perimeter reduces |(r)| near walls lowers transverse mode pumping R moves toward T 0
• Axial contour (along x): smooth thermal profile reduces |/x| detunes longitudinal mode frequencies and/or lowers |R|
• Throat/nozzle (boundaries): tune impedance (resonators/dampers/temporal resistors) disrupt in-phase (, u) at outlet R 0 T 0

Practical settings:

• Wall belt: section spacing 1020R around circumference; T_wall minimal (just enough for effect), monitor |T|/T and avoid thermal stress
• Axial contour: 23 independent heating/cooling zones along length; tune for minimum |f/f| and |R|
• Boundary impedance: combine -wave and Helmholtz resonators with temporal resistors (elements generating counter-phase (, u))

Target control metric: maintain R = ()"u T 0 (see Diagram 7.A)

(3) Pilots (Local Ignition/Stabilizing Jets) Goal: create local anchors in phase opposite to target mode to flip R sign locally (in pumping hotspots)

(4) Temporal Resistors Absorptive/phase-shifting inserts designed for target mode frequency to generate counter-phase

• Implementation: cavity + thermally regulated insert/layer with specified (t) response
• Concept: analogous to acoustic absorbers, but designed in -language

Diagram 7.D Control Lever Tree (Local R and Gradient Type)

Код

R(x) > 0 ?

?

No? yes which gradient dominates?

? No? axial (/x) T(x) profiles, staggering q

? "? radial (/r) wall cooling, pilots

? ?

? "? if insufficient locally temporal resistor

"? no monitor (maintain R (C) 0)

Node explanations:

• Axial gradient (/x): affects longitudinal mode frequency shift (see 5.2, Fig. H) and reduces |R| via detuning
• Radial gradient (/r): drives transverse mode splitting (see 5.1, Fig. G); wall thermal belts and pilots reduce |(r)| in sensitive zones
• Temporal resistor: element/setup generating counter-phase (, u) at boundary or locally (-language impedance analog)
• In any branch recompute local R(x) after intervention and aim for integral goal: R T 0

Practical notes:

• When selecting frequency and phase of intervention, target the mode with highest R contribution (via R(x) map)
• For multi-frequency regimes MIMO setup: multiple independent staggering/pilot channels, optimized for R vector across modes

7.3. Integration with Resonators/Dampers (Classical + TT View)

Classical devices are not discarded TTВН provides a clear temporal interpretation and integration rules

• Helmholtz resonators: pressure oscillation suction interpreted as local minimum and counter-phase near wall synergizes with || smoothing
• Quarter-wave dampers: act as phase shifters for temporal perturbations in target frequency band

8. Applicability Limits and Discussion

8.1. Where TTВН Adds Value and Where It Matches Classical Theory

Areas of Significant Advantage of TTВН

• Interpretation of phase relationships: TTВН provides an ontological foundation for the Rayleigh criterion: in-phase p and q in-phase and u, and the sign of "c'"()"u explains why energy pumping occurs.
• New predictions: Linear and verifiable effects: splitting of degenerate transverse modes under radial (r), and frequency shift of longitudinal modes under axial /x.
• Unified control framework: A single target metric R = ()"u enables joint optimization of classical tools (dampers, resonators) and temporal levers (staggering q, smoothing ||, temporal resistors).
• Causality map: Clear separation: sources (q ), transport ( u), accumulation (E_mode), losses (D).

Areas of Equivalence with Classical Theory

• Conservation laws for energy and momentum: Remain unchanged; TTВН simply repackages sources and flows in -language.
• Quantitative thresholds for self-excitation: With proper calibration of (T, p, s), thresholds and growth rates match classical estimates.
• Modal dynamics in linear approximation: Growth rates and mode maps are essentially the same; the difference lies in the convenience of using R as a unified metric instead of multiple phase conditions.

Diagram 8.A Where TTВН Adds Value vs. Matches Classical Theory

Код

Added Value

New Predictions (split, f) """""

Unified Metric R """"

Phase & Gradient Control """

------------------------------------------ Classical Theory

Conservation Laws "

Threshold Estimates "

Modal Frequencies (Linear) "

Reader Notes:

• New predictions transverse mode splitting (split) and axial frequency shift (f), see 5
• Unified metric R general criterion for pumping/anti-pumping, see 7.1
• Phase and gradient control practical levers (staggering, thermal smoothing, impedance)
• Classical equivalence conservation laws, basic thresholds, linear modal frequencies

8.2. Sensitivity to a_s, a_T, a_p and Calibration Protocol

First-Order Sensitivity Hierarchy:

• Temperature coefficient a_T critical: Due to large relative gradients |T|/T, sensitivity to a_T is highest. Target relative calibration accuracy: 1010' for reliable f predictions.
• Pressure coefficient a_p moderate importance: Requires ~10' accuracy for reproducible frequency shifts and stability maps.
• Entropy coefficient a_s secondary in fast linear analysis: Can be temporarily neglected for screening, but becomes important with significant composition/evaporation changes.

Minimal Calibration Protocol:

1. Static ( calibration): Measure T(x,r), p(x,r) in steady-state; fit (a_T, a_p, optionally a_s) so that - A_T"ln T + A_p"ln p matches n(x) fields (via n , T conversion)
2. Dynamic ( verification): Synchronously measure p, q (PLIF/CH*), u (LDV/PIV), Schlieren; check sign of "c'"()"u and its correlation with p"q dt
3. Cross-validation: Repeat on different geometry/regime (vary L_eff, R, composition); confirm stability of a_T, a_p within declared accuracy

Diagram 8.B Calibration Pipeline

Код

Static: T, p

fit: (a_T, a_p [, a_s])

(x,r),

sanity check (Schlieren)

Dynamic: p, q, u, Schlieren

,

R = ()"u

compare with p"q dt

Cross-validation: new geometry/regime

coefficient stability

Error Notes:

• R is most affected by: phase misalignment between channels, u noise, numerical differentiation of
• Targets: phase accuracy (C) 35R, spatial resolution (C) 1 mm in hot zones of gradients

8.3. Common Misinterpretations (No Energy from Time)

Critical Clarifications:

• Energy is not created by time: The field is not an energy source; it parameterizes how existing chemical energy via q is converted into mode energy.
• Temporal gradients are a descriptive language: is derived from thermodynamic fields T, p, s via = (T,p,s) and their gradients not a new extra energy field.
• Causality and conservation laws are preserved: Mass, momentum, and energy balances hold in standard form; TTВН simply shifts emphasis to phase synchronization of and u.

Common Misconceptions and Responses:

• Time creates energy No. Energy comes from chemical source (q); "c'"()"u is a form of energy redistribution
• Temporal force violates thermodynamics No. f_T = "c'" is an equivalent expression of inertial/gravitational effects in chosen parameterization; conservation laws remain intact
• Effects arise from nothing No. arises from measurable gradients of T, p, s; TTВН predictions depend on calibrated but physical coefficients a_s, a_T, a_p

Diagram 8.C Energy Flow (No Mysticism)

Код

Chemical Energy

q(t)

T, p, s

(T, p, s)

u

E_mode

Losses D (viscosity, conduction, radiation, nozzle radiation)

Energy Balance:

dE_mode/dt = "c'"()"u D

(conservation laws remain standard)

8.4. Applicability Boundaries and When to Extend the Model

• Strong mean flows / shear layers: Must explicitly account for convection and mode advection (u"), and boundary impedance in -language
• Strong nonlinearity: Large amplitudes generate harmonics and subharmonics; requires weakly nonlinear balance extension and possibly modal reduction with interaction terms
• Composition and phase transitions: With significant variation and evaporation, a_s becomes important requires separate calibration via s and reaction delays
• Complex boundary conditions: Permeable walls, active injectors, resonators: surface terms from integration by parts yield boundary pumping must be explicitly included

Diagram 8.D When to Extend the Model

Код

R > 0 under idealizations?

add convection and boundaries ( impedance)

Large amplitudes?

include weakly nonlinear terms ( harmonics)

Composition / evaporation?

calibrate a_s and chemical delays

Mismatch in f / split?

check a_T, a_p calibration and phase synchronization

Section 8 Summary

1. TTВН adds most value where phase and gradients matter: explaining energy pumping, new predictions (split, f), unified control metric R
2. For basic conservation laws and linear thresholds, TTВН matches classical theory when (T,p,s) is properly calibrated
3. Misinterpretations are resolved by remembering: is a parameterization of thermal fields not a new energy source
4. For complex regimes, extended model is used: convection, boundaries, weak nonlinearity, composition effects

9. Conclusion

9.1. Summary

The Temporal Theory of High-Frequency Instability (TTВН) provides a coherent platform linking time dynamics (rate and potential ) to oscillatory processes in combustion chambers. Based on principles from the Temporal Theory of Gravity, it constructs a mathematical framework demonstrating the equivalence between temporal energy pumping and the classical Rayleigh criterion:

• The work of temporal force f_T = "c'" over a period, W_T "c'" ()"u dt, has the same sign as p"q dt
• The formalism does not replace thermoacoustics but deepens it, revealing the root cause of modal energy feeding through in-phase and u

The theory leads to falsifiable predictions:

(a) Splitting of degenerate transverse modes under radial gradient (r) (b) Frequency shift of longitudinal modes under axial gradient /x

It also introduces a control metric: R = ()"u as the target quantity

9.2. Practical Takeaways

1. Target metric: In design and tuning, aim for R T 0 (anti-pumping)
2. Active control: Manage phase and gradients staggering q(t), local pilots, distributed wall heating/cooling (smoothing ||), temporal resistors
3. Hybrid suppression: Optimize resonators/dampers, reinterpreting them as boundary -impedance tools
4. Diagnostics: Deploy proxy measurement systems for and (Schlieren/Shimmer, PLIF/CH*, p sensors, PIV/LDV) with synchronization for R evaluation

Diagram 9.A Closed-Loop Control via R

Код

Diagnostics: p, q, u, Schlieren Reconstruction of

R = ( " u)

'???????????????????????????????

? ?

if R > 0 if R T 0

(pumping/growth) (anti-pumping)

?

adjust q phase (staggering) monitor and maintain

smooth ||, insert resistor

9.3. Implementation Roadmap (Minimal Cycle)

Step 1. Calibrate (T, p, s): fit a_T, a_p[, a_s] under steady-state conditions Step 2. Verify linear predictions: measure f under imposed /x and split under /r; confirm linear trends and sign Step 3. Introduce R metric (online/offline) and select control levers Step 4. Close the loop on R: achieve R T 0, assess change in growth rate and spectrum Step 5. Integrate with classical tools: tune resonators/dampers for temporal phase; perform A/B comparison of regimes

Diagram 9.B Implementation Roadmap

Код

Calibrate

Predictions (f, split)

Metric R

R loop closure

Hybrid with dampers

(Static) (Linear verification) (Online) (Active control) (Optimization)

9.4. Future Directions

• Metamaterials for control: Wall/insert structures with tailored (t) response for target frequencies
• Distributed thermal management: Programmable T(x,r) profiles for adaptive /x and /r
• Real-time R control: Fast R estimators and MIMO algorithms for staggering/pilots
• Expanded object class: Solid/hybrid propellant engines, turbines, heat extraction chambers, plasma sources
• Deep verification: Multi-geometry, scaling, varied compositions and regimes; publish full error budgets

9.5. Limitations and Reduction Risks (Honest Framework)

• Not energy from time: is a parameterization of thermal fields, not a new energy source; conservation laws remain intact
• Calibration sensitivity: Errors in a_T, a_p[, a_s] and phase synchronization distort R
• Linearity: Sections 25 are valid to first order; large amplitudes require weakly nonlinear analysis and convection/boundary effects
• Risk of reducing everything to : In engineering, -formalism must complement not replace hydrodynamics and chemistry, or key mechanisms may be lost

9.6. Final Statement

TTВН offers a fundamentally motivated yet operationally measurable description of high-frequency instability. It:

• Explains the root cause of modal energy feeding via
• Provides new, testable predictions (split, f)
• Defines a unified control metric (R T 0) and practical levers to achieve it
• Integrates naturally with classical suppression methods

This opens the path to a new generation of stable energy systems, where regime control is guided not only by pressure and heat, but by the properties of time as an active physical substance precisely calibrated and reliably measured.

10. Recommended Literature for TTВН

I. Classical Foundations of Thermoacoustics and Combustion Instability

II. Advanced Diagnostics and Control Methods

III. Foundational Works on the Nature of Time (Ontological Basis)

IV. Specialized Works on High-Frequency Instability in LREs

V. Signal Processing and Data Analysis (Experimental Section)

Citation Strategy

1. Introduction and Section 3: Actively cite Rayleigh (1), Lieuwen (2), Culick (3) shows TTВН builds on a solid foundation
2. Section 2 (Ontology): Cite Einstein (8) and Landau-Lifshitz (9) adds physical weight to the concept of time as substance
3. Sections 56 (Experiment): Kohse-Hinghaus (4), Raffel (5), Bendat (12) demonstrate mastery of modern experimental methods
4. Appendices DE: Use Harrje (10) for calibration examples and scale estimates

Appendix A. Derivation of Pumping Equivalence (Rayleigh Formalism)

A.0. Objective

Demonstrate the equivalence between the sign of temporal energy pumping and the Rayleigh criterion:

W_T = " c' " ( " u) dt p(t) " q(t) dt > 0 condition for mode growth (Rayleigh)

A.1. Definitions and Notation

• Temporal force (per unit volume):f_T = " c' "
• Work over period P in volume V:W_T = _V f_T " u dV dt

A.2. Integration by Parts (Volume Boundaries)

Using the identity for divergence:

"("u) = ()"u + "("u) ()"u = "("u) "("u)

Integrating over volume V and time t:

V ( " u) dV dt = V "(u"n) dS dt _V "("u) dV dt

Assuming rigid or weakly permeable walls (u"n - 0) and small at boundaries, the surface term is negligible:

_V ( " u) dV dt - _V "("u) dV dt

A.3. Mass Continuity (Linear Form)

Linearized continuity equation:

/t + "("u) - 0 "u - (1/)"(/t)

Substituting into previous expression:

_V ( " u) dV dt - (1/)" _V "(/t) dV dt

Thus:

W_T - ("c' / )" _V "(/t) dV dt

A.4. Phenomenological Link Between and Thermodynamic Variables

First-order relation:

- a_s"s + a_T"(T/T) + a_p"(p/p)

For an ideal gas (linear perturbations):

s - c_p"(T/T) R"(p/p)

Therefore:

- A_T"(T/T) + A_p"(p/p) whereA_T = a_T + a_s"c_p,A_p = a_p a_s"R

A.5. Energy Equation and Phase Coupling (Link to q)

Linearized energy equation (first approximation):

"T"(ds/dt) - q

Thus:

s q q (considering phase delays from chemical kinetics and mixing)

Combined with linear relations between , p, T, s, we find:

If _V p"q dV dt > 0 then"c'" _V ( " u) dV dt > 0

The signs match (mode growth)

A.6. Sign Equivalence (Final Criterion)

Under linear conditions, rigid walls, and proper calibration of (T, p, s):

"c'" _V ( " u) dV dt > 0 _V p"q dV dt > 0

In other words, temporal pumping is equivalent to the Rayleigh criterion in terms of work sign.

A.7. Sinusoidal Example (Sign Check)

Assume sinusoidal perturbations at a point (or mode):

p(t) = P"cos("t) q(t) = Q"cos("t q) u(t) = "sin("t) = "cos("t 90R) (t) = "cos("t )

Then over period P = 2/:

p"q dt (P"Q / 2)"cos(q) ( " u) dt (" / 2)"cos( 90R) = (" / 2)"sin(_)

If experiment/model shows mode growth (Rayleigh):

p"q dt > 0cos(_q) > 0_q (90R, +90R)

In -language, this corresponds to:

"c'" ( " u) dt > 0sin(_) < 0 (assuming local orientation of u)

Both criteria yield the same sign for growth (with consistent phase and direction conventions)

A.8. Assumptions and Applicability Limits

• Linear regime:|X| |X| (validity of modal superposition)
• Boundaries:Rigid or weakly permeable; for active/porous boundaries, surface terms must be included yields boundary pumping in -formulation
• Thermodynamics:Ideal gas as reference; real mixtures require calibration of a_s, a_T, a_p
• Losses:Viscosity, conduction, radiation reduce growth/decay magnitude but not sign equivalence
• Mean flows:Significant u requires accounting for convection and mode advection proper reduction preserves correspondence

A.9. Pseudographic Correspondence Diagram

Код

q(t,x) T, p, s (T,p,s) (t,x) u(t,x) W_T

? ? ? ?

"? Rayleigh: p"q dt ????????????????%? TTВН: "c'" ("u) dt ?...

Sign equivalence (growth / decay)

A.10. Conclusion

The temporal formulation of energy pumping:

W_T = "c'" _V ( " u) dV dt

is equivalent to the Rayleigh criterion:

_V p " q dV dt > 0

in terms of work sign over one period, assuming proper calibration of (T, p, s) and validity of linear assumptions.

Appendix B. Pseudographic Schemes AH (Monospaced Font)

Usage Notes:

• Schemes AC Section 2 (Ontology and Estimates)
• Schemes DF Section 3 (Phasing and Rayleigh Equivalence)
• Schemes GH Section 5 (Predictions)
• Keep monospaced font and disable line wrapping when transferring to .docx

Scheme 14.A Field and Vectors (Section 2.1)

Код

'?????????????? Chamber ????????????????

? ?    ? ? center hot higher

? ? ? ? walls cooler lower

? "    ў ? points from fast zones to slow

? ?

? ?

"?????????????????????????????????????...

Scheme 15.B TTВН Force Chain (Section 2.2)

Код

T, p, s (thermo fields)

?

= a"s + a"ln T + a"ln p phenomenological link

?

(temporal potential)

?

(gradient of time rate)

?

f = " c' " temporal force per volume

?

W " c' " ( " u) dt energy pumping into mode

Scheme 16.C Estimating || from Pumping (Section 2.4)

Код

p ~ 0.5 MPa

L ~ 0.5 m

~ 1 kg/m

c ~ 3"10 m/s

|| - p / ( " c' " L) - 10 m

Scheme 17.D Phasing of p and q (Section 3.1)

Код

p(t): ??/\??/\??/\??

q(t): ?/\??/\??/\?

in-phase p " q dt > 0 mode growth

Scheme 18.E -Language Equivalent (Section 3.2)

Код

:

u:

( " u):

R = ( " u) < 0

W " c' " R > 0 mode growth

Scheme 19.F Rayleigh TTВН Correspondence (Section 3.3 / Appendix A)

Код

q(t,x) T, p, s (T,p,s) (t,x) u(t,x) W

? ? ? ?

"?? Rayleigh: p " q dt ????????%? TTВН: " c' " ( " u) dt ?...

sign equivalence (growth / decay)

Scheme 20.G Mode Splitting under (r) (Section 5.1)

Код

Transverse mode (before):

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With (r) 0 splitting:

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split |/r| splitting amplitude depends on radial gradient

Scheme 21.H Frequency Shift under /x (Section 5.2)

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Longitudinal mode (before):f = f_base

With /x 0 shift:

f f + f

f /x sign and magnitude depend on axial gradient

Appendix C. Symbol and Unit Table (SI)

Notes

1. Normalization in - a_s"s + a_T"ln(T/T) + a_p"ln(p/p): Logarithms are taken over dimensionless arguments. If entropy s is not normalized, then [a_s] = (J"kg"K). It is convenient to define normalized entropy s = sc_p then a_s becomes dimensionless. In reduced form: - A_T"ln(T/T) + A_p"ln(p/p)
2. Sign and dimensionality of R_: R_ has units of s, and the temporal power per unit volume is: P_T = "c'"R_[W"m]

Convention:

-R_ < 0P_T > 0energy is pumped into the mode (growth, suppression required) -R_ T 0P_T (C) 0anti-pumping or neutral (stable)

Appendix D. Order-of-Magnitude Estimates for Prototype LREs

D.1. Assumptions and Ranges

• Regime: steady-state operation, linear oscillation domain (small perturbations)
• Gas: hot combustion products, - 0.52.0 kg"m, T - 25003600 K
• Geometry: cylindrical chamber + nozzle; characteristic length L - 0.30.8 m
• Upper modal frequencies of interest: f_max - 10100 kHz
• Pressure oscillation amplitudes: p - 0.11.0 MPa
• Oscillatory velocity: |u| - 220 m"s
• Sign conventions: R_ = ()"uP_T = "c'"R_ R_ < 0 P_T > 0 energy pumping (mode growth) R_ T 0 stabilization / neutral

D.2. Two Models for Estimating ||

• Model A (via required force pumping): || - p / ("c'"L)
• Model B (via thermodynamic gradients and (T,p,s)): - A_T"ln(T/T) + A_p"ln(p/p) || |A_T|"|T|/T + |A_p|"|p|/p

Typical gradients: |T|/T ~ 10 m|p|/p ~ 10'10 m A_T, A_p dimensionless calibration coefficients (typically 10'1)

D.3. Numerical Estimates for Two Prototypes

D.3.1. Prototype 1 Laboratory-Scale Chamber

• p - 1.5 MPaT - 3000 K - 1.0 kg"m
• L - 0.30 mp - 0.5 MPa|u| - 510 m"s
• |T|/T - 10 m|p|/p - 10'10 m
• Calibration: A_T - 0.05A_p - 0.01

Estimates:

• || (Model A): - 510 / (1"(310)'"0.30) - 1.8510 m
• || (Model B): - 0.0510 + 0.01(10'10) 50 + 110 logarithmic scale yields - 10'10 m practical intersection: ~10 m
• R_:- (1010)"(510) 510110 s
• P_T:- "c'"R_ - 910"(510110) 4.510910 W"m

Frequency effects:

• Longitudinal modes (/x):f/f ~ 0.32 %
• Transverse modes (/r):split/f ~ 0.21.5 %

D.3.2. Prototype 2 Full-Scale Mid-Class Chamber

• p - 812 MPaT - 3200 K - 0.81.5 kg"m
• L - 0.600.80 mp - 0.31.0 MPa|u| - 515 m"s
• A_T - 0.020.10A_p - 0.0050.05

Estimates:

• || (Model A): - (310110) / (910"0.7) (4.815)10' m
• || (Model B):~10'10 m
• R_:- (510'110)"(515) 2.5101.510 s
• P_T:- (0.81.5)"910"(2.5101.510) 1.8102.010 W"m

Frequency effects:

• f/f (longitudinal):~0.23 %
• split/f (transverse):~0.12 %

D.4. Summary Table Quick Reference

D.5. Sign Interpretation and Practical Scale

• R_ - 10 s or lower high likelihood of mode growth (pumping)
• R_ ~ 1010 s sensitive zone; adjust phase (staggering), smooth ||, tune impedance
• R_ T 0 target suppression regime (anti-pumping / neutral)

Traffic Light Reminder: R_ < 0 interveneR_ = 0 neutralR_ > 0 stabilize (see Fig. 7.A)

D.6. Diagnostic Requirements (from Estimates)

• Temporal resolution:t (C) 10 s (for modes up to ~100 kHz)
• Spatial resolution:(C) 1 mm in zones of peak
• Sampling rate for p, accelerometers:T

Appendix E. Experimental Protocol Template (TTВН)

Purpose: Ensure reproducibility of R_, , phasing, and frequency effect measurements in combustion chambers. The protocol covers setup, synchronization, diagnostics, processing, and reporting.

? E.1. General Information

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Test rig name: ____________________________

Chamber type: Laboratory Full-scale Other: ____________

Gas medium: ____________________________

Date / time of experiment: ____________________________

Responsible engineer: ____________________________

? E.2. Operating Conditions

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Chamber pressure (p): ____________ MPa

Temperature (T): ____________ K

Density (): ____________ kg"m

Chamber length (L): ____________ m

Pressure oscillation (p): ____________ MPa

Oscillatory velocity (|u|): ____________ m"s

? E.3. (T, p, s) Calibration

Код

Coefficients:

a_s = ____________

a_T = ____________

a_p = ____________

Reduced form:

A_T = a_T + a_s"c_p = ____________

A_p = a_p a_s"R = ____________

Reference values:

T = ____________ K

p = ____________ Pa

? E.4. Diagnostic Channels

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p sensors (sampling rate): ____________ kHz

PLIF / CH* (q mapping): Yes No

Schlieren / shimmer (n ): Yes No

LDV / PIV (u): LDV PIV Both

Thermocouples / IR (T, T): Yes No

Accelerometers / wall sensors: Yes No

? E.5. Synchronization and Phasing

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Reference channel: p u q Other: ____________

Sync method: Hardware trigger Cross-correlation Markers

Phase accuracy: ____________ R

SNR (key channels): ____________ dB

? E.6. Measured Parameters

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Gradients:

/x = ____________ m

|/r| = ____________ m

Pumping metric:

R_ = ( " u) = ____________ s

P_T = "c'"R_ = ____________ W"m

Frequency effects:

f = ____________ Hz

f = ____________ Hz

f/f = ____________ %

split = ____________ Hz

split/f = ____________ %

? E.7. Processing and Interpretation

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estimation model:

A (via p / "c'"L)

B (via thermal gradients)

R_ interpretation:

R_ < 0 pumping

R_ - 0 neutral

R_ > 0 anti-pumping

Applied control levers:

q staggering

thermal contouring (heating/cooling)

pilot jets

temporal resistors

resonators / dampers

? E.8. Uncertainties and Conditions

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Temporal resolution: ____________ s

Spatial resolution: ____________ mm

Phase error: ____________ R

uncertainty: ____________ %

Notes: ____________________________________________